Abstract
We develop the homological theory of KLR algebras of symmetric affine type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.
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1 Introduction
Khovanov–Lauda–Rouquier algebras (henceforth KLR algebras), also known as Quiver Hecke algebras, are a family of \({\mathbb {Z}}\)-graded associative algebras introduced by Khovanov and Lauda [11] and Rouquier [23] for the purposes of categorifying quantum groups. More specifically they categorify the upper-triangular part \({\mathbf {f}}=U_q({\mathfrak {g}})^+\) of the quantised enveloping algebra of a symmetrisable Kac–Moody Lie algebra \({\mathfrak {g}}\)—see Sect. 5 for a precise statement. Let I be the set of simple roots of \({\mathfrak {g}}\) and \({\mathbb {N}}I\) the monoid of formal sums of elements of I. For each \(\nu \in {\mathbb {N}}I\) there is an associated KLR algebra \(R(\nu )\).
In this paper we will assume that \({\mathfrak {g}}\) is of symmetric affine type. For now however, we will describe the theory developed in [4, 22] where \({\mathfrak {g}}\) is finite dimensional. The results of this paper generalise these results to the symmetric affine case.
One begins with choosing a convex order \(\prec \) on the set of positive roots satisfying a convexity property—see Definition 3.1. It is this convex order which determines a PBW basis of \({\mathbf {f}}\). The representation theory of KLR algebras is built via induction functors from the theory of cuspidal representations. Write \(\{\alpha _1\succ \cdots \succ \alpha _N\}\) for the set of positive roots, remembering that we are temporarily discussing the finite type case.
To each root \(\alpha \) there is a subcategory of \(R(\alpha )\)-modules which are cuspidal defined in Definition 8.3. There is a unique irreducible cuspidal module \(L(\alpha )\). Let \(\Delta (\alpha )\) be the projective cover of \(L(\alpha )\) in the category of cuspidal \(R(\alpha )\)-modules.
Given any sequence \(\pi =(\pi _1,\ldots ,\pi _N)\) of natural numbers, the proper standard and standard modules are defined respectively by
where \(\circ \) denotes the induction of a tensor product and \((\pi _i)\) is a divided power construction. Then in [22] it is proved that the modules \(\overline{\Delta }(\pi )\) categorify the dual PBW basis, have a unique irreducible quotient and that these quotients give a classification of all irreducible modules. In [4] it is proved that the modules \(\Delta (\pi )\) categorify the PBW basis and their homological properties are studied, justifying the use of the term standard.
Now let us turn our attention to the results of this paper where \({\mathfrak {g}}\) is of symmetric affine type. Again the starting point is the choice of a convex order \(\prec \) on the set of positive roots. The theory of PBW bases for affine quantised enveloping algebras dates back to the work of Beck [3] and is considerably more complicated than the theory in finite type. It is a feature of the literature that the theory of PBW bases is only developed for convex orders of a particular form. We rectify this problem by presenting a construction of PBW bases in full generality.
For \(\alpha \) a real root, the category of cuspidal \(R(\alpha )\)-modules is again equivalent to the category of k[z]-modules while the category of semicuspidal \(R(n\alpha )\)-modules is again equivalent to modules over a polynomial algebra. Whereas in finite type the proofs of these results currently rest on some case by case computations, here we give a uniform proof, the cornerstone of which is the growth estimates in Sect. 15.
For the imaginary roots, the category of semicuspidal representations is qualitatively very different. The key observation here is that the R-matrices constructed by Kang, Kashiwara and Kim [9] enable us to determine an isomorphism
where M is either an irreducible cuspidal \(R(\delta )\)-module or an indecomposable projective in the category of cuspidal \(R(\delta )\)-modules (here \(\delta \) is the minimal imaginary root). We are then able to use the representation theory of the symmetric group to decompose these modules \(M^{\circ n}\). This presence of the symmetric group as an endomorphism algebra can also be seen to explain the appearance of Schur functions in the definition of a PBW basis in affine type.
With the semicuspidal modules understood we are able to prove our main theorems which are analogous to those discussed above in finite type. Namely families of proper standard and standard modules are constructed which categorify the dual PBW and PBW bases respectively. See Theorem 24.4 and the following paragraph for this result. Compared with the corresponding theorem in [13, Proposition 4.11], we are able to identify the imaginary constituents of the PBW basis. The proper standard modules have a unique irreducible quotient which gives a classification of all irreducibles and the standard modules satisfy homological properties befitting their name, leading to a BGG reciprocity theorem.
As a consequence we obtain a new positivity result, Theorem 24.10, which states that when an element of the canonical basis of \({\mathbf {f}}\) is expanded in a PBW basis, the coefficients that appear are polynomials in q and \(q^{-1}\) with non-negative coefficients (and the transition matrix is unitriangular).
We thank A. Kleshchev, P. Tingley and B. Webster for useful conversations.
2 Preliminaries
The purpose of this section is to collect standard notation about root systems and other objects which we will be making use of in this paper.
Let \((I,\cdot )\) be a Cartan Datum of symmetric affine type. Following the approach of Lusztig [19], this comprises a finite set I and a symmetric pairing \(\cdot \,{:}\,I\times I\!\rightarrow \!{\mathbb {Z}}\) such that \(i\cdot i=2\) for all \(i\in I\), \(i\cdot j\le 0\) if \(i\ne j\) and the matrix \((i\cdot j)_{i,j\in I}\) is of corank 1. Such Cartan data are completely classified and correspond to the extended Dynkin diagrams of type A, D and E. We extend \(\cdot \,{:}\,I\times I\!\rightarrow \!{\mathbb {Z}}\) to a bilinear pairing \({\mathbb {N}}I\times {\mathbb {N}}I\rightarrow {\mathbb {Z}}\).
Let \(\Phi ^+\) be the set of positive roots in the corresponding root system. We identify I with the set of simple roots of \(\Phi ^+\). In this way we are able to meaningfully talk about elements of \({\mathbb {N}}I\) as being roots.
The set of real roots of \(\Phi ^+\) is denoted \(\Phi ^+_{re}\).
For \(\nu =\sum _{i\in I}\nu _i\cdot i\in I\), define \(|\nu |=\sum _{i\in I }\nu _i\). If \(\nu \) happens to be a root, we also call this the height of the root and denote it \({\text {ht}}(\nu )\).
Let \(\Phi _f\) be the underlying finite type root system. A chamber coweight is a fundamental coweight for some choice of positive system on \(\Phi _f\). If a positive system is given, let \(\Omega \) denote the set of chamber coweights with respect to this system.
Let \(p\,{:}\,\Phi \!\rightarrow \!\Phi _{f}\) denote the projection from the affine root system to the finite root system whose kernel is spanned by the minimal imaginary root \(\delta \). For \(\alpha \in \Phi _f\), let \(\tilde{\alpha }\) denote the minimal positive root in \(p^{-1}(\alpha )\).
Let \(W=\langle s_i\mid i\in I\rangle \) be the Weyl group of \(\Phi \), generated by the simple reflection \(s_i\) which is the reflection in the hyperplane perpendicular to \(\alpha _i\).
Let \(\Delta _f\) be the standard set of simple roots in \(\Phi _f\). Let \(W_f\) be the finite Weyl group.
Let \({\mathcal {P}}\) denote the set of partitions. A multipartition \({\underline{\lambda }}=\{\lambda _\omega \}_{\omega \in \Omega }\) is a sequence of partitions indexed by \(\Omega \). We write \({\underline{\lambda }}\vdash n\) if \(\sum _\omega |\lambda _\omega |=n\).
The symmetric group on n letters is denoted \(S_n\). If \(\mu ,\nu \in I\), the element \(w[\mu ,\nu ]\in S_{|\mu +\nu |}\) is defined by
3 Convex orders on root systems
Definition 3.1
A convex order on \(\Phi ^+\) is a total preorder \(\preceq \) on \(\Phi ^+\) such that
-
If \(\alpha \preceq \beta \) and \(\alpha +\beta \) is a root, then \(\alpha \preceq \alpha +\beta \preceq \beta \).
-
If \(\alpha \preceq \beta \) and \(\beta \preceq \alpha \) then \(\alpha \) and \(\beta \) are imaginary roots.
Theorem 3.2
A convex order \(\prec \) on \(\Phi ^+\) satisfies the following condition:
-
Suppose A and B are disjoint subsets of \(\Phi ^+\) such that \(\alpha \prec \beta \) for any \(\alpha \in A\) and \(\beta \in B\). Then the cones formed by the \({\mathbb {R}}_{\ge 0}\) spans of A and B meet only at the origin.
Remark 3.3
In [25], this condition replaces our first condition in their definition of a convex order. This theorem shows that their definition and our definition agree.
Remark 3.4
The following proof requires being in finite or affine type since it depends on the positive semidefiniteness of the natural bilinear form. We do not know if a similar statement is possible for more general root systems.
Proof
We will write \((\cdot ,\cdot )\) for the natural bilinear form on the root lattice. Let \(\{\alpha _i\}\) be a finite set of roots in A and let \(\{b_j\}\) be a finite set of roots in B. For want of a contradiction, suppose that for some positive real numbers \(c_i,d_j\) we have
Let
Then \((c_1,c_2,\ldots ,d_1,d_2,\ldots )\in W\otimes _{\mathbb {Q}}{\mathbb {R}}\), since the root system \(\Phi \) is defined over \({\mathbb {Q}}\). As \({\mathbb {Q}}\) is dense in \({\mathbb {R}}\), W is dense in \(W\otimes _{\mathbb {Q}}{\mathbb {R}}\). So there is a point in W with all coordinates positive. Hence we can assume that each \(c_i\) and \(d_j\) are rational numbers without loss of generality. Clearing denominators, we can assume they lie in \({\mathbb {Z}}\).
Now suppose we have a solution to (3.1) where the \(c_i\) and \(d_j\) are positive integers with \(\sum _i c_i + \sum _j d_j\) as small as possible.
For any \(i\ne j\), if \(\alpha _i+\alpha _j\) were a root, we could replace one occurrence of \(\alpha _i\) and \(\alpha _j\) by the single root \(\alpha _i+\alpha _j\) to get a smaller solution, contradicting our minimality assumption. Therefore \(\alpha _i+\alpha _j\) is not a root for any \(i\ne j\). This implies that \((\alpha _i,\alpha _j)\ge 0\) for \(i\ne j\).
If all \(\alpha _i\) and \(\beta _j\) are imaginary, there is a contradiction since there is only one imaginary direction. So there exists at least one real root in the equation we are studying, without loss of generality say it is \(\alpha _k\).
Applying \((\cdot ,\alpha _k)\) leaves us with the inequality
Therefore there exists j such that \((\beta _j,\alpha _k)>0\), which implies that \(\beta _j-\alpha _k\) is a positive root. By convexity this root must be greater than \(\beta _j\). So now we may subtract \(\alpha _k\) from both sides of (3.1) to obtain a smaller solution, again contradicting minimality.
Therefore no solution to (3.1) can exist, as required. \(\square \)
The imaginary roots are all multiples of a fundamental imaginary root, which we will denote \(\delta \). In any convex order, these imaginary roots must all be equal to each other.
Let \(\prec \) be a convex order. The set of positive real roots is divided into two disjoint subsets, namely
and
If we can write \(\Phi _{\prec \delta } = \{ \alpha _1 \prec \alpha _2\prec \cdots \}\) and \(\Phi _{\succ \delta } = \{ \beta _1 \succ \beta _2\succ \cdots \}\) for some sequences of roots \(\{\alpha _i\}_{i=1}^\infty \) and \(\{\beta _j\}_{j=1}^\infty \), then we say that \(\prec \) is of word type.
Example 3.5
Let \((V,\le )\) be a totally ordered \({\mathbb {Q}}\)-vector space. Let \(h\,{:}\,{\mathbb {Q}}\Phi \!\rightarrow \!V\) be an injective linear transformation. For two positive roots \(\alpha \) and \(\beta \), say that \(\alpha \prec \beta \) if \(h(\alpha )/{\text {ht}}(\alpha )<h(\beta )/{\text {ht}}(\beta )\) and \(\alpha \preceq \beta \) if \(h(\alpha )/{\text {ht}}(\alpha )\le h(\beta )/{\text {ht}}(\beta )\). This defines a convex order on \(\Phi \).
In the above example, we can take \(V={\mathbb {R}}\) with the standard ordering to get the existence of many convex orders of word type.
An example of a convex order not of word type which we will make use of later on is the following:
Example 3.6
Let \(V={\mathbb {R}}^2\) where \((x,y)\le (x',y')\) if \(x<x'\) or \(x=x'\) and \(y\le y'\). Let \(\Delta _f\) be a simple system in \(\Phi _f\) and pick \(\alpha \in \Delta _f\). Define \(h\,{:}\,{\mathbb {Q}}\Phi \!\rightarrow \!V\) by \(h(\tilde{\alpha })=(0,1)\), \(h(\tilde{\beta })=(x_\beta ,0)\) for \(\beta \in \Delta _f{\setminus }\{\alpha \}\) where the \(x_\beta \) are generically chosen positive real numbers, and \(h(\delta )=0\). We extend by linearity, noting that \(\{\delta \}\cup \{\tilde{\beta }\mid \beta \in \Delta _f\}\) is a basis of \({\mathbb {Q}}\Phi \).
In this example, we have
and all other positive roots are either greater than \(\widetilde{\alpha }\) or less than \(\widetilde{-\alpha }\).
Recall that p is the projection from the affine root system to the finite root system.
Lemma 3.7
There exists \(w\in W_f\) such that \(p(\Phi _{\prec \delta })=w \Phi ^+_f\) and \(p(\Phi _{\succ \delta }) = w \Phi ^-_f\).
Proof
First suppose that \(\alpha \in p(\Phi _{\prec \delta })\) and \(-\alpha \in p(\Phi _{\prec \delta })\). Then there are integers m and n such that the affine roots \(-\alpha +m\delta \) and \(\alpha +n\delta \) are both less than \(\delta \) in the convex order \(\prec \). By convexity, their sum \((m+n)\delta \) is also less that \(\delta \), a contradiction. Since a similar argument holds for \(p(\Phi _{\succ \delta })\), we see that for each finite root \(\alpha \), exactly one of \(\alpha \) and \(-\alpha \) lies in \(p(\Phi _{\prec \delta })\).
Now suppose that \(\alpha ,\beta \in p(\Phi _{\prec \delta })\) and \(\alpha +\beta \) is a root. Then for some integers m and n, the affine roots \(\alpha +m\delta \) and \(\beta +n\delta \) are both less than \(\delta \). By convexity, their sum \((\alpha +\beta )+(m+n)\delta \), which is also an affine root, is also less than \(\delta \). Therefore \(\alpha +\beta \in p(\Phi _{\prec \delta })\).
We have shown that \(p(\Phi _{\prec \delta })\) is a positive system in the finite root system \(\Phi _f\). This suffices to prove the lemma. \(\square \)
Define a finite initial segment to be a finite set of roots \(\alpha _1\prec \alpha _2\prec \cdots \prec \alpha _N\) such that for all positive roots \(\beta \), either \(\beta \succ \alpha _i\) for all \(i=1,\ldots ,N\) or \(\beta =\alpha _i\) for some i.
For any \(w\in W\) define \(\Phi (w)=\{\alpha \in \Phi ^+\mid w^{-1}\alpha \in \Phi ^-\}\).
Lemma 3.8
Let \(\alpha _1\prec \alpha _2 \prec \cdots \prec \alpha _N\) be a finite initial segment. Then there exists \(w\in W \) such that \(\{ \alpha _1,\ldots \alpha _N \} =\Phi (w)\). Furthermore there exists a reduced expression \(w=s_{i_1}\cdots s_{i_N}\) such that \(\alpha _k=s_{i_1}\cdots s_{i_{k-1}} \alpha _{i_k}\) for \(k=1,\ldots N\).
Proof
The proof proceeds by induction on N. For the base case where \(N=1\), any root \(\alpha \) which is not simple is the sum of two roots \(\alpha =\beta +\gamma \). By convexity of \(\prec \), either \(\beta \prec \alpha \prec \gamma \) or \(\gamma \prec \alpha \prec \beta \). Either way, \(\alpha \ne \alpha _1\) so \(\alpha _1\) is simple, \(\alpha _1=\alpha _i\) for some \(i\in I\) and we take \(w=s_i\).
Now assume that the result is known for initial segments with fewer than N roots. Let \(v=s_{i_1}\ldots s_{i_{N-1}}\). Consider \(v^{-1}\alpha _N\). By inductive hypothesis, it is a positive root. Suppose for want of a contradiction that \(v^{-1}\alpha _N\) is not simple. Then we can find positive roots \(\beta \) and \(\gamma \) such that \(v^{-1}\alpha _N=\beta +\gamma \).
We can’t have \(v\beta =\alpha _N\) as this would force \(\gamma =0\). If \(v\beta =\alpha _j\) for some \(j<N\) then \(\beta =v^{-1}\alpha _j\) which by inductive hypothesis is in \(\Phi ^-\), a contradiction. Therefore either \(v\beta \) is a positive root satisfying \(v\beta \succ \alpha _N\) or \(v\beta \in \Phi ^-\). A similar statement holds for \(v\gamma \).
To have both \(v\beta \) and \(v\gamma \) greater than \(\alpha _N\) contradicts the convexity of \(\prec \). Therefore, without loss of generality, we may assume \(v\beta \in \Phi ^-\). Then \(-v\beta \) is a positive root with \(v^{-1}(-v\beta )=-\beta \) which is a negative root, so by inductive hypothesis, \(-v\beta =\alpha _j\) for some \(j<N\). Now consider the equation \(\alpha _N + (-v\beta )=v\gamma \). The convexity of \(\prec \) implies that \(v\gamma =\alpha _{j'}\) for some \(j'<N\). This option is shown to be impossible in the previous paragraph, creating a contradiction. Therefore \(v^{-1}\alpha _N\) must be a simple root.
Define \(i_N\in I\) by \(\alpha _{i_N}=v^{-1}\alpha _N\) and let \(w=s_{i_1}\cdots s_{i_N}\). It remains to show that \(\{ \alpha _1,\ldots \alpha _N \} = \{\alpha \in \Phi ^+ \mid w^{-1}\alpha \in \Phi ^-\}\).
If \(\beta \) is a positive root that is not equal to \(\alpha _j\) for some \(j\le N\), then by inductive hypothesis \(v^{-1}\beta \in \Phi ^+\). Then \(w^{-1}\beta =s_{i_N}(v^{-1}\beta )\in \Phi ^-\) if and only if \(v^{-1}\beta =\alpha _{i_N}\) which isn’t the case since this is equivalent to \(\beta =\alpha _N\).
If \(\beta =\alpha _j\) for some \(j<N\) then \(v^{-1}\beta \in \Phi ^-\). So \(w^{-1}\beta =s_{i_N}(v^{-1}\beta )\in \Phi ^-\) unless \(v^{-1}\beta =-\alpha _{i_N}\). This isn’t the case since it is equivalent to \(\beta =-\alpha _{i_N}\).
The above two paragraphs show that for a positive root \(\beta \), if \(\beta \in \{\alpha _1,\ldots ,\alpha _{N-1}\}\) then \(w^{-1}\beta \in \Phi ^+\) while if \(\beta \notin \{\alpha _1,\ldots ,\alpha _N\}\), then \(w^{-1}\beta \in \Phi ^+\). Since \(w^{-1}\alpha _N=-\alpha _{i_N}\in \Phi ^-\), this completes the proof. \(\square \)
Lemma 3.9
[7] The restriction of a convex order to \(\Phi _{\prec \delta }\) is of the form
for some positive integer n.
For a convex order \(\prec \), define
Lemma 3.10
Let \(\prec \) be a convex order. Let \(\beta \) be the smallest root that is not in any initial segment of \(\Phi ^+\). Assume that \(\beta \) is real. Let S be a finite set of roots containing \(\beta \). Then there exists a convex order \(\prec '\) such that \(I(\prec ')=I(\prec )\cup \{\beta \}\) and the restrictions of \(\prec \) and \(\prec '\) to S are the same.
Proof
Let L be the set of roots in \(\Phi ^+\) less than or equal to \(\beta \) under \(\prec \). Then by [6, Theorem 3.12], there exists \(v,t\in W\) with t a translation and \(L=\cup _{n=1}^\infty \Phi (vt^n)\).
Let w be such that \(S\subset \{\alpha _1\prec \cdots \prec \alpha _N\}=\Phi _w\). There exists an integer n such that \(\Phi (w)\cup \{\beta \}\subset \Phi (vt^n)\). Let \(v'=vt^n\). Since \(\Phi (v')\supset \Phi (w)\), for any reduced expression of w, there exists a reduced expression of \(v'\) beginning with that of w.
We choose the reduced decomposition of w to be compatible with \(\prec \). Then extend the reduced decomposition as per the above to get a new ordering \(\prec '\) on L. This has the desired properties. \(\square \)
Theorem 3.11
Let S be a finite subset of \(\Phi ^+\) and let \(\prec \) be a convex order on \(\Phi ^+\). Then there exists a convex order \(\prec '\) of word type such that the restrictions of \(\prec \) and \(\prec '\) to S are equivalent.
Proof
Suppose our convex order begins
and that \(S\cap \{\alpha _i\mid i\in {\mathbb {Z}}^+\} \subset \{\alpha _1,\ldots ,\alpha _n\}\). We now define inductively a sequence of convex orders \(\prec _i\) with \(I(\prec _i)=\{\alpha _i\mid i\in {\mathbb {Z}}^+\}\cup \{\beta _1,\ldots ,\beta _i\}\) as follows:
Set \(\prec _0=\prec \). Assume that \(\prec _{i}\) is constructed. To construct \(\prec _{i+1}\), apply Lemma 3.10 with \(S=\{\alpha _1,\ldots ,\alpha _{n+i},\beta _1,\ldots ,\beta _{i}\}\). We will take the convex order denoted \(\prec '\) whose existence is given to us by Lemma 3.10 as \(\prec _{i+1}\).
Now let \(\prec ''=\lim _{i\rightarrow \infty } \prec _i\). If \(\prec \) is of n-row type, then \(\prec ''\) will be of \((n-1)\)-row type and the restrictions of \(\prec \) and \(\prec ''\) to S are the same. After iterating this process we reach a new convex order \(\prec '\) whose restriction to S is the same as \(\prec \) and is of word type on \(\Phi _{\prec \delta }\). Repeating this construction on the set of roots greater than \(\delta \) completes the proof of this theorem. \(\square \)
Remark 3.12
Using this theorem it will often be possible to assume without loss of generality that the convex order \(\prec \) is of word type.
4 The algebra \({\mathbf {f}}\)
The algebra \({\mathbf {f}}_{{\mathbb {Q}}(q)}\) is the \({\mathbb {Q}}(q)\) algebra as defined in [19] generated by elements \(\{\theta _i\mid i\in I\}\). Lusztig defines it as the quotient of a free algebra by the radical of a bilinear form. By the quantum Gabber–Kac theorem, it can also be defined in terms of the Serre relations. Morally, \({\mathbf {f}}_{{\mathbb {Q}}(q)}\) should be thought of as the positive part of the quantised enveloping algebra \(U_q({\mathfrak {g}})\). There is only a slight difference in the coproduct, necessary as the coproduct in \(U_q({\mathfrak {g}})\) does not map \(U_q({\mathfrak {g}})^+\) into \(U_q({\mathfrak {g}})^+\otimes U_q({\mathfrak {g}})^+\).
There is a \({\mathbb {Z}}[q,q^{-1}]\)-form of \({\mathbf {f}}_{{\mathbb {Q}}(q)}\), which we denote simply by \({\mathbf {f}}\). It is the \({\mathbb {Z}}[q,q^{-1}]\)-subalgebra of \({\mathbf {f}}_{{\mathbb {Q}}(q)}\) generated by the divided powers \(\theta _i^{(n)}=\theta _i^n/[n]_q!\), where \([n]_q\pi !=\prod _{i=1}^n(q^i-q^{-i})/(q-q^{-1})\) is the q-factorial. If \({\mathcal {A}}\) is any \({\mathbb {Z}}[q,q^{-1}]\)-algebra, we use the notation \({\mathbf {f}}_{\mathcal {A}}\) for \({\mathcal {A}}\otimes _{{\mathbb {Z}}[q,q^{-1}]}{\mathbf {f}}\).
The algebra \({\mathbf {f}}\) is graded by \({\mathbb {N}}I\) where \(\theta _i\) has degree i for all \(i\in I\). We write \({\mathbf {f}}=\oplus _{\nu \in {\mathbb {N}}I} {\mathbf {f}}_\nu \) for its decomposition into graded components. Of significant importance for us is the dimension formula
The tensor product \({\mathbf {f}}\otimes {\mathbf {f}}\) has an algebra structure given by
where \(y_1\) and \(x_2\) are homogeneous of degree \(\beta _1\) and \(\alpha _2\) respectively.
Given a bilinear form \((\cdot ,\cdot )\) on \({\mathbf {f}}\), we obtain a bilinear form \((\cdot ,\cdot )\) on \({\mathbf {f}}\otimes {\mathbf {f}}\) by
There is a unique algebra homomorphism \(r\,{:}\,{\mathbf {f}}\!\rightarrow \!{\mathbf {f}}\otimes {\mathbf {f}}\) such that \(r(\theta _i)=\theta _i\otimes 1 + 1 \otimes \theta _i\) for all \(i\in I\).
The algebra \({\mathbf {f}}\) has a symmetric bilinear form \(\langle \cdot ,\cdot \rangle \) satisfying
The form \(\langle \cdot ,\cdot \rangle \) is nondegenerate. Indeed, in the definition of \({\mathbf {f}}\) in [19], \({\mathbf {f}}\) is defined to be the quotient of a free algebra by the radical of this bilinear form. It is known that \({\mathbf {f}}\) is a free \({\mathbb {Z}}[q,q^{-1}]\)-module. Let \({\mathbf {f}}^*\) be the graded dual of \({\mathbf {f}}\) with respect to \(\langle \cdot ,\cdot \rangle \). By definition, \({\mathbf {f}}^*=\oplus _{\nu \in {\mathbb {N}}I} {\mathbf {f}}_\nu ^*\). As twisted bialgebras over \({\mathbb {Q}}(q)\), \({\mathbf {f}}_{{\mathbb {Q}}(q)}\) and \({\mathbf {f}}^*_{{\mathbb {Q}}(q)}\) are isomorphic, though there is no such isomorphism between their integral forms.
5 KLR algebras
A good introduction to the basic theory of KLR algebras appears in [16, §4]. Although it is not customary, we will first give the geometric construction of KLR algebras, then discuss the standard presentation in terms of generators and relations. In this paper, we must restrict ourselves to KLR algebras which come from geometry. The primary reason for this restriction is our reliance on the theory of R-matrices, which we introduce in Sect. 14. The results presented in Sect. 7 also require the geometric interpretation.
Define a graph with vertex set I and with \(-i\cdot j\) edges between i and j for all \(i\ne j\). Let Q be the quiver obtained by placing an orientation on this graph.
For \(\nu \in {\mathbb {N}}I\), define \(E_\nu \) and \(G_\nu \) by
With the obvious action of \(G_\nu \) on \(E_\nu \), \(E_\nu /G_\nu \) is the moduli stack of representations of Q with dimension vector \(\nu \).
Let \({\mathcal {F}}_\nu \) be the complex variety whose points consist of a point of \(E_\nu \), together with a full flag of subrepresentations of the corresponding representation of Q. The variety \({\mathcal {F}}_\nu \) is a disjoint union of smooth connected varieties. Let \({\mathcal {F}}_\nu =\sqcup _\mathbf{i}{\mathcal {F}}_\nu ^\mathbf{i}\) be its decomposition into connected components and \(\pi ^\mathbf{i}\,{:}\,{\mathcal {F}}_\nu ^\mathbf{i}\!\rightarrow \!E_\nu \) be the natural \(G_\nu \)-equivariant morphism. Define
For each \(\nu \in {\mathbb {N}}I\) we define the KLR algebra \(R(\nu )\) by
We now introduce the more customary approach via generators and relations. This presentation is due to [26] and [23], and more recently over \({\mathbb {Z}}\) in [21]. To introduce this presentation, we first need to define, for any \(\nu \in {\mathbb {N}}I\),
This is acted upon by the symmetric group \(S_{|\nu |}\) in which the adjacent transposition \((i,i+1)\) is denoted \(s_i\).
Define the polynomials \(Q_{i,j}(u,v)\) for \(i,j\in I\) by
where the products are over the sets of edges from i to j and from j to i, respectively.
Theorem 5.1
The KLR algebra \(R(\nu )\) is the associative \({\mathbb {Q}}\)-algebra generated by elements \(e_\mathbf{i}\), \(y_j\), \(\tau _k\) with \(\mathbf{i}\in \,\hbox {Seq}\,(\nu )\), \(1\le j\le |\nu |\) and \(1\le k< |\nu |\) subject to the relations
Remark 5.2
Although the polynomials \(Q_{i,j}(u,v)\) are not exactly as they appear in [11], the reader should not be concerned when we quote results from [11] as all of the arguments go through without change. The discussion in [12] shows that changing the ordering of the quiver Q does not change the isomorphism type of \(R(\nu )\).
The KLR algebras \(R(\nu )\) are \({\mathbb {Z}}\)-graded, where \(e_{\mathbf{i}}\) is of degree zero, \(y_j e_\mathbf{i}\) is of degree \(\mathbf{i}_j \cdot \mathbf{i}_j\) and \(\phi _k e_\mathbf{i}\) is of degree \(-\mathbf{i}_k \cdot \mathbf{i}_{k+1}\).
They satisfy the property that \(R(\nu )_d=0\) for d sufficiently negative (depending on \(\nu \)) and \(R(\nu )_d\) is finite dimensional for all d. Relevant implications of these properties are that there are a finite number of isomorphism classes of simple modules and that projective covers exist.
All representations of KLR algebras that we consider will be finitely generated \({\mathbb {Z}}\)-graded representations. If needed, we write \(M=\oplus _d M_d\) for the decomposition of a module M into graded pieces. A submodule of a finitely generated \(R(\nu )\)-module is finitely generated by [11, Corollary 2.11].
For a module M, we denote its grading shift by i by \(q^iM\), this is the module with \((q^i M)_d = M_{d- i}\).
Given two modules M and N, we consider \({\text {Hom}}(M,N)\), and more generally \({\text {Ext}}^i(M,N)\) as graded vector spaces. All Ext groups which appear in the paper will be taken in the category of \(R(\nu )\)-modules.
Let \(\tau \) be the antiautomorphism of \(R(\nu )\) which is the identity on all generators \(e_\mathbf{i},y_i,\phi _j\). For any \(R(\nu )\)-module M, there is a dual module \(M^\circledast = {\text {Hom}}_{\mathbb {Q}}(M,{\mathbb {Q}})\), where the \(R(\nu )\) action is given by \((x\lambda )(m)=\lambda (\tau (x)m)\) for all \(x\in R(\nu )\), \(\lambda \in M^\circledast \) and \(m\in M\).
For every irreducible \(R(\nu )\)-module L, there is a unique choice of grading shift such that \(L^\circledast \cong L\) [11].
Let \(\lambda ,\mu \in {\mathbb {N}}I\). Then there is a natural inclusion \(\iota _{\lambda ,\mu }:R(\lambda )\otimes R(\mu )\rightarrow R(\lambda +\mu )\), defined by \(\iota _{\lambda ,\mu }(e_\mathbf{i}\otimes e_\mathbf{j})=e_{\mathbf{i}\mathbf{j}}\), \(\iota _{\lambda ,\mu }(y_i\otimes 1)=y_i\), \(\iota _{\lambda ,\mu }(1\otimes y_i)=y_{i+|\lambda |}\), \(\iota _{\lambda ,\mu }(\phi _i\otimes 1)=\phi _i\), \(\iota _{\lambda ,\mu }(1\otimes \phi _i)=\phi _{i+|\lambda |}\).
Define the induction functor \({\text {Ind}}_{\lambda ,\mu }:R(\lambda )\otimes R(\mu )\hbox {-mod}\rightarrow R(\lambda +\mu )\hbox {-mod}\) by
Define the restriction functor \({\text {Res}}_{\lambda ,\mu }:R(\lambda +\mu )\hbox {-mod}\rightarrow R(\lambda )\otimes R(\mu )\hbox {-mod}\) by
The induction and restriction functors are both exact.
For a \(R(\lambda )\)-module A and a \(R(\mu )\)-module B, we write \(A\circ B\) for \({\text {Ind}}_{\lambda ,\mu }(A\otimes B)\). Under duality, the behaviour is
Khovanov and Lauda [11] prove the existence of a dual pair of isomorphisms
and
The category \(R(\nu )\hbox {-pmod}\) is the category of finitely generated projective \(R(\nu )\)-modules and \(G_0\) means to take the split Grothendieck group. The category \(R(\nu )\hbox {-fmod}\) is the category of finite dimensional \(R(\nu )\)-modules and \(K_0\) means to take the Grothendieck group. We denote the class of a module M, identified with its image under the above isomorphisms, by [M]. The action of \(q\in {\mathcal {A}}\) is by grading shift.
The functors of induction and restriction decategorify to a product and coproduct. The isomorphisms above are then isomorphisms of twisted bialgebras.
If M is a general finitely generated \(R(\nu )\)-module, then it has a well-defined composition series, where each composition factor appears with a multiplicity that is an element of \({\mathbb {Z}}((q))\). Thus we can consider [M] to be an element of \({\mathbf {f}}^*_{{\mathbb {Z}}((q))}\).
Of great importance will be the following Mackey theorem. The general case stated below has the same proof as the special case presented in [11].
Theorem 5.3
[11, Proposition 2.18] Let \(\lambda _1,\ldots ,\lambda _k,\mu _1\ldots ,\mu _l\in {\mathbb {N}}I\) be such that \(\sum _i \lambda _i=\sum _j \mu _j\) and let M be a \(R(\lambda _1)\otimes \cdots \otimes R(\lambda _k)\)-module. Then the module \( {\text {Res}}_{\mu _1,\ldots ,\mu _l}\circ {\text {Ind}}_{\lambda _1,\ldots ,\lambda _k}(M) \) has a filtration indexed by tuples \(\nu _{ij}\) satisfying \(\lambda _i=\sum _j \nu _{ij}\) and \(\mu _j=\sum _i\nu _{ij}\). The subquotients of this filtration are isomorphic, up to a grading shift, to the composition \({\text {Ind}}_\nu ^\mu \circ \tau \circ {\text {Res}}_\nu ^\lambda (M)\). Here \({\text {Res}}_{\nu }^\lambda \,{:}\,\otimes _i R(\lambda _i)\hbox {-mod}\!\rightarrow \!\otimes _i(\otimes _jR(\nu _{ij}))\hbox {-mod}\) is the tensor product of the \({\text {Res}}_{\nu _{i\bullet }}\), \(\tau \,{:}\,\otimes _i(\otimes _jR(\nu _{ij}))\hbox {-mod}\!\rightarrow \!\otimes _j(\otimes _iR(\nu _{ij}))\hbox {-mod}\) is given by permuting the tensor factors and \({\text {Ind}}_\nu ^\mu \,{:}\,\otimes _j(\otimes _iR(\nu _{ij}))\hbox {-mod}\!\rightarrow \!\otimes _j R(\mu _j)\hbox {-mod}\) is the tensor product of the \({\text {Ind}}_{\nu _{\bullet i}}\).
We refer to the filtration appearing in the above theorem as the Mackey filtration. It will be very common for us to make arguments using vanishing properties of modules under restriction to greatly restrict the number of these subquotients which can be nonzero.
For each \(w\in S_{|\nu |}\), make a choice of a reduced decomposition \(w=s_1\ldots s_n\) as a product of simple reflections. Define \(\tau _w=\tau _{1}\cdots \tau _{n}\). In general \(\tau _w\) depends on the choice of reduced decomposition though this is not the case for permutations of the form \(w[\beta ,\gamma ]\).
Theorem 5.4
[11, Theorem 2.5][23, Theorem 3.7] The set of elements of the form \(y_1^{a_1}\cdots y_{|\nu |}^{a_{|\nu |}}\tau _w e_\mathbf{i}\) with \(a_1,\ldots ,a_{|\nu |}\in {\mathbb {N}}\), \(w\in S_{|\nu |}\) and \(\mathbf{i}\in \,\hbox {Seq}\,(\nu )\) is a basis of \(R(\nu )\).
Theorem 5.5
[11, Corollary 2.11] The KLR algebra is Noetherian.
We work over the ground field \({\mathbb {Q}}\). It is proved in [11] that any irreducible module is absolutely irreducible, so there is no change to the theory in passing to a field extension. This also means that any irreducible module for a tensor product of KLR algebras is a tensor product of irreducibles, a fact we use without comment.
6 Adjunctions
In addition to the induction and restriction functor defined in the previous section, there is also a coinduction functor \({\text {CoInd}}_{\lambda ,\mu }R(\lambda )\otimes R(\mu )\hbox {-mod}\rightarrow R(\lambda +\mu )\hbox {-mod}\), defined by
where the \(R(\lambda +\mu )\) module structure on \({\text {CoInd}}_{\lambda ,\mu }(M)\) is given by \((rf)(t)=f(tr)\) for \(f\in {\text {CoInd}}_{\lambda ,\mu }(M)\) and \(r,t\in R(\lambda +\mu )\).
The following adjunctions are standard:
Proposition 6.1
The functor \({\text {Ind}}_{\lambda ,\mu }\) is left adjoint to \({\text {Res}}_{\lambda ,\mu }\), while the functor \({\text {CoInd}}_{\lambda ,\mu }\) is right adjoint to \({\text {Res}}_{\lambda ,\mu }\).
As a \(R(\lambda )\otimes R(\mu )\) module, \(R(\lambda +\mu )\) is free of finite rank. This implies that the induction, restriction and coinduction functors all send projective modules to projective modules. As a consequence, there are natural isomorphisms of higher Ext groups
for all \(A\in R(\lambda )\hbox {-mod}\), \(B\in R(\mu )\hbox {-mod}\) and \(C\in R(\lambda +\mu )\hbox {-mod}\).
Let \(\sigma _\nu \,{:}\,R(\nu )\!\rightarrow \!R(\nu )\) be the involutive isomorphism of \(R(\nu )\) with \(\sigma _\nu (e_\mathbf{i})=e_{w_0\mathbf{i}}\), \(\sigma _\nu (y_i)=y_{|\nu |+1-i}\) and \(\sigma _\nu (\tau _j e_\mathbf{i})=(1-2\delta _{\mathbf{i}_j,\mathbf{i}_{j+1}})\tau _{|\nu |-j}e_{w_0\mathbf{i}}\). This induces an autoequivalence \(\sigma _\nu ^*\) of \(R(\nu )\hbox {-mod}\).
Theorem 6.2
[20, Theorem 2.2] There is a natural equivalence of functors
Proof
The statement of this theorem in [20] includes a hypothesis that the modules in question are all finite dimensional. Exactly the same proof works for graded modules all of whose pieces are finite dimensional, which covers all the modules we will ever come across. The general case follows by writing a module as the direct limit of its finitely generated submodules (noting that \(R(\lambda +\mu )\) is finite over \(R(\lambda )\otimes R(\mu )\)). \(\square \)
Remark 6.3
Most importantly, applied to a module of the form \(A\otimes B\) yields an isomorphism
In particular, there is an isomorphism
for any \(R(\lambda )\)-module C, \(R(\mu )\)-module B and \(R(\lambda +\mu )\)-module A.
There are parabolic analogues of all the functors and results discussed in this section.
7 The Ext bilinear form
By the decomposition theorem [1], we have
where each \(L_b\) is a nonzero finite dimensional graded vector space and \({\mathcal {P}}_b\) is an irreducible \(G_\nu \)-equivariant perverse sheaf on \(E_\nu \). The indexing set \({\mathcal {B}}_\nu \) can be taken to be the set of elements of weight \(\nu \) in the crystal \({\mathcal {{\mathcal {B}}}}(\infty )\), though for our purposes it is not necessary to know this fact.
The maximal semisimple quotient of \(R(\nu )\) is \(\oplus _{b\in {\mathcal {B}}_\nu } {\text {End}}(L_b)\) and hence the simple representations of \(R(\nu )\) are the multiplicity spaces \(L_b\). The projective cover of \(L_b\) is the module \(\oplus _{d\in {\mathbb {Z}}}{\text {Hom}}_{D^b_{G_\nu }(E_\nu )}({\mathcal {L}},{\mathcal {P}}_b[d])\). In this way we get a bijection between simple perverse summands of \(\pi _!\underline{{\mathbb {Q}}}\) and irreducible representations of \(R(\nu )\). By Lusztig’s geometric construction of canonical bases, the class of a simple representation under the isomorphism (5.4) lies in the dual canonical basis while the class of its projective cover under (5.3) lies in the canonical basis.
As has been noted by Kato [8], each algebra \(R(\nu )\) is graded Morita equivalent to the algebra
The algebra \(A(\nu )\) is a \({\mathbb {N}}\)-graded algebra with \(A(\nu )_0\) semisimple. Under this Morita equivalence the self-dual irreducible module \(L_b\) gets sent to a one-dimensional representation of \(A(\nu )\) concentrated in degree zero.
Lemma 7.1
Let M be a finitely generated representation of \(R(\nu )\) and let N be a finite dimensional representation of \(R(\nu )\). Fix an integer d. Then there exists \(i_0\) such that \({\text {Ext}}^i(M,N)_d=0\) for all \(i>i_0\).
Proof
Replace \(R(\nu )\) with the Morita equivalent algebra \(A(\nu )\) and assume that M and N are \(A(\nu )\)-modules. Let \(\cdots \rightarrow P^1\rightarrow P^0 \rightarrow M\rightarrow 0\) be a minimal projective resolution of M. As M is finitely generated, there exists \(d_0\) such that \(M_j=0\) for \(j<d_0\). Since \(A(\nu )\) is nonnegatively graded with \(A(\nu )_0\) semisimple, \(P^i_j=0\) for \(j<d_0+i\). The vector space \({\text {Ext}}^i(M,N)\) is a subquotient of \({\text {Hom}}(P^i,N)\) and for sufficiently large i, \({\text {Hom}}(P^i,N)_d=0\) by degree considerations. \(\square \)
By the above lemma, if M is a finitely generated \(R(\nu )\)-module and N is a finite dimensional \(R(\nu )\)-module, then the infinite sum
is a well-defined element of \({\mathbb {Z}}((q))\). We thus get a pairing on Grothendieck groups
Lemma 7.2
The pairing \((\cdot ,\cdot )\) satisfies the following properties
for all \(x,y,z\in {\mathbf {f}}\), \(f(q)\in {\mathbb {Z}}((q))\), \({\mathfrak {g}}(q)\in {\mathbb {Z}}[q,q^{-1}]\), where y and z are homogeneous of degree \(\beta \) and \(\gamma \).
Proof
The first formula is obvious. The second is a simple computation in \(R(i)\cong k[z]\). The third follows from (6.1) and the fourth follows from (6.2). \(\square \)
Let \(\langle x,y\rangle =( x,\bar{y})\). The pairing \(\langle \cdot ,\cdot \rangle \) can be extended by \({\mathbb {Z}}((q))\)-linearity to give a bilinear pairing on \({\mathbf {f}}^*_{{\mathbb {Z}}((q))}\).
Lemma 7.3
The pairing \(\langle \cdot ,\cdot \rangle \) satisfies the following properties
Proof
These follow from the analogous formulae in Lemma 7.2. To derive the third we need to know that r commutes with the bar involution while to derive the fourth we need to know that \(\overline{yz}=q^{\beta \cdot \gamma }\bar{z}\bar{y}\) for homogeneous elements y and z of degree \(\beta \) and \(\gamma \). \(\square \)
Corollary 7.4
The pairing \(\langle \cdot ,\cdot \rangle \) defined using the Ext-pairing is equal to the usual pairing on \({\mathbf {f}}\) as in the end of Sect. 4.
Proof
It is immediate that there is a unique pairing satisfying the properties of Lemma 7.3 and these properties define the pairing in [19]. \(\square \)
Lemma 7.5
Let M be a finite dimensional \(R(\nu )\)-module with
where the second sum is over all self-dual simple modules L. If \(a_{n,L}\ne 0\) then \(q^n L\) is a submodule of M while if \(a_{m,L}\ne 0\) then \(q^m L\) is a quotient of M.
Proof
If this lemma is false, then there exist self-dual irreducible representations \(L_1\) and \(L_2\) of \(R(\nu )\), and an integer \(d\le 0\) such that \({\text {Ext}}^1(L_1,L_2)_d\ne 0\). Now replace \(R(\nu )\) by the Morita equivalent \(A(\nu )\). We compute \({\text {Ext}}^1(L_1,L_2)\) by computing a minimal projective resolution of \(L_1\). Since \(A(\nu )\) is non-negatively graded with \(A(\nu )_0\) semisimple, we see from this computation that \({\text {Ext}}^1(L_1,L_2)\) is concentrated in degrees greater than zero. \(\square \)
8 Proper standard modules
Definition 8.1
Let \(\alpha \) be a positive root and n be an integer. A representation L of \(R(n\alpha )\) is called semicuspidal if \({\text {Res}}_{\lambda ,\mu }L\ne 0\) implies that \(\lambda \) is a sum of roots less than or equal to \(\alpha \) and \(\mu \) is a sum of roots greater than or equal to \(\alpha \).
Lemma 8.2
Let \(\alpha \) be a positive root, \(m_1,\ldots ,m_n\in {\mathbb {Z}}^+\) and \(L_i\) be a semicuspidal representation of \(R(m_i\alpha )\) for each \(i=1,2,\ldots ,n\). Then the module \(L_1\circ \cdots \circ L_n\) is semicuspidal.
Proof
This immediate from Theorem 5.3 and the definition of semicuspidality. \(\square \)
Definition 8.3
Let \(\alpha \) be a positive root. A representation L of \(R(\alpha )\) is called cuspidal if whenever \({\text {Res}}_{\lambda ,\mu }L\ne 0\) and \(\lambda ,\mu \ne 0\), we have that \(\lambda \) is a sum of roots less than \(\alpha \) and \(\mu \) is a sum of roots greater than \(\alpha \).
Remark 8.4
It is clear that if \(\alpha \) is an indivisible root then any semicuspidal representation of \(R(\alpha )\) is cuspidal. If \(\alpha =n\delta \) for \(n\ge 2\) we will see in Theorem 19.10 that there are no cuspidal representations of \(R(\alpha )\).
Definition 8.5
A sequence of modules \(L_1,\ldots ,L_n\) is called admissible if each \(L_i\) is an irreducible semicuspidal representation of \(R(m_i\alpha _i)\) with \(m_i\in {\mathbb {Z}}^+\) and the positive roots \(\alpha _i\) satisfy \(\alpha _1\succ \alpha _2\succ \cdots \succ \alpha _n\).
Lemma 8.6
Let \(\alpha _1\succ \alpha _2\succ \cdots \succ \alpha _k\) and \(\beta _1\succ \beta _2\succ \cdots \succ \beta _l\) be positive roots and \(m_1,\ldots ,m_k\), \(n_1,\ldots ,n_l\) be positive integers. Let \(L_1,\ldots ,L_k\) be semicuspidal representations of \(R(m_1\alpha _1),\cdots ,R(m_k\alpha _k)\) respectively. Then
where we are considering bilexicographical ordering on the multisets \(\underline{\alpha }\) and \(\underline{\beta }\).
Proof
Consider a nonzero layer of the Mackey filtration for \({\text {Res}}_{n_1\beta _1,\ldots ,n_l\beta _l} L_1\circ \cdots \circ L_n \). It is indexed by a set of elements \(\nu _{ij}\in {\mathbb {N}}I\) such that \(m_i\alpha _i=\sum _j \nu _{ij}\) and \(n_j\beta _j = \sum _{i}\nu _{ij}\). For the piece of the filtration to be nonzero, it must be that \({\text {Res}}_{\nu _{i,1},\ldots ,\nu _{i,n}}L_i \ne 0\) for each i.
Suppose that t is an index such that \(m_i\alpha _i=n_i\beta _i\) for \(i<t\). We will prove that in order for us to have a nonzero piece of the filtration, it must be that either \(\beta _t\prec \alpha _t\) or \(m_t\alpha _t=n_t\beta _t=\nu _{t,t}\).
By induction on t, we may assume that \(\nu _{ii}=m_i\alpha _i=n_i\beta _i\) for \(i<t\). Therefore \(\nu _{i,j}=0\) for all i and j with \(i\ge t\) and \(j<t\).
Suppose \(i\ge t\). Since the module \(L_i\) is cuspidal, this implies that \(\nu _{i,t}\) is a sum of roots less than or equal to \(\alpha _i\), which are all less than or equal to \(\alpha _t\).
Now \(n_t\beta _t=\sum _{i\ge t}\nu _{i,t}\) is written as a sum of positive roots all less than or equal to \(\alpha _t\). Therefore, by convexity of the ordering, either \(\beta _t\prec \alpha _t\) or \(n_t\beta _t=m_t\alpha _t\). In this latter case, equality in our inequalities must hold everywhere, hence \(\nu _{t,t}=n_t\beta _t\) as required.
This is enough to conclude that \(\underline{\alpha }\ge \underline{\beta }\) under lexicographical ordering. Similarly we get \(\underline{\alpha }\ge \underline{\beta }\) under reverse lexicographical ordering, so we have \(\underline{\alpha }\ge \underline{\beta }\) under the bilexicographical order.\(\square \)
Lemma 8.7
Let \(\alpha _1\succ \alpha _2\succ \cdots \succ \alpha _n\) be roots, \(m_1,\ldots ,m_n\) be positive integers and \(L_1,\ldots ,L_n\) be irreducible semicuspidal representations of \(R(m_1\alpha _1),\ldots ,R(m_n\alpha _n)\) respectively. Then
-
(1)
the module \(L_1\circ \cdots \circ L_n\) has a unique irreducible quotient L, and
-
(2)
\({\text {Res}}_{m_1\alpha _1,\ldots ,m_n\alpha _n} L_1\circ \cdots \circ L_n = {\text {Res}}_{m_1\alpha _1,\ldots ,m_n\alpha _n}L=L_1\otimes \cdots \otimes L_n\).
Proof
Suppose that Q is a nonzero quotient of \(L_1\circ \cdots \circ L_n\). Then by adjunction there is a nonzero map from \(L_1\otimes \cdots \otimes L_n\) to \({\text {Res}}Q\). As \(L_1\otimes \cdots \otimes L_n\) is irreducible, this map is injective.
The restriction functor is exact and by Lemma 8.6, \({\text {Res}}(L_1\circ \cdots \circ L_n)\) is simple. Therefore the head of \(L_1\circ \cdots \circ L_n\) must be simple. \(\square \)
If \(L_1,\ldots ,L_n\) is a sequence of representations, we define \(A(L_1,\ldots ,L_n)={\text {cosoc}}(L_1\circ \cdots L_n)\). The below two theorems appear also in [13] and [25]. We provide proofs of both after the statement of Theorem 8.9.
Theorem 8.8
Every irreducible module for \(R(\nu )\) is of the form \(A(L_1,\ldots ,L_n)\) for exactly one set of irreducible semicuspidal representations \(L_1,\ldots ,L_n\) of \(R(m_1\alpha _1),\ldots ,R(m_n\alpha _n)\) respectively, where \(\alpha _1\succ \cdots \succ \alpha _n\) are positive roots.
Theorem 8.9
If \(\alpha \) is a positive real root and n is a positive integer, there is one simple semicuspidal module for \(R(n\alpha )\). For the imaginary roots, let f(n) be the number of simple semicuspidal representations of \(R(n\delta )\) (and set \(f(0)=1\)). Then
Proof
Here we prove Theorems 8.8 and 8.9 by a simultaneous induction on \(\nu \).
First let us consider the case where \(\nu \) is not of the form \(n\alpha \) for some root \(\alpha \). The number of irreducible representations of \(R(\nu )\) is equal to \(\dim {{\mathbf {f}}_\nu }\), which is the coefficient of \(t^\nu \) in the power series (4.1).
By inductive hypothesis applied to Theorem 8.9, the number of admissible sequences of semicuspidal modules \((L_1,\ldots ,L_n)\) is equal to \(\dim {\mathbf {f}}_\nu \). By Lemma 8.7, each of the modules \(A(L_1,\ldots ,L_n)\) are irreducible, and by applying various restriction functors, we see via Lemma 8.6 that these modules are all distinct. Therefore we have identified all of the irreducible \(R(\nu )\)-modules in this case, proving Theorem 8.8.
Now we turn our attention to the case where \(\nu =k\alpha \) for some root \(\alpha \). By the same arguments as in the previous case, the modules of the form \(A(L_1,\ldots ,L_n)\) where \(n\ge 2\) yield all the irreducible modules for \(R(k\alpha )\) except one, unless \(\nu =n\delta \), when the construction yields all irreducible modules except f(n). It suffices to prove that if L is an irreducible representation of \(R(\nu )\) with L not of the form \(A(L_1,\ldots ,L_n)\) with \(n\ge 2\), then L is semicuspidal.
Suppose that \(\lambda \) and \(\mu \) are such that \({\text {Res}}_{\lambda \mu }L\ne 0\). We need to prove that \(\lambda \) is a sum of roots less than or equal to \(\alpha \) (the result for \(\mu \) is similar) and we may suppose that neither of \(\lambda \) and \(\mu \) is zero. Let \(L_\lambda \otimes L_\mu \) be an irreducible submodule of \({\text {Res}}_{\lambda \mu }L\). By inductive hypothesis \(L_\lambda =A(L_1,\ldots ,L_k)\) for some admissible sequence of semicuspidal representations. Suppose that \(L_1\) is a \(R(m\beta )\) module where \(\beta \) is a root. Then \({\text {Res}}_{m\beta ,\nu -m\beta }L\ne 0\). If \(\beta \preceq \alpha \), then \(\lambda \) is a sum of roots less than or equal to \(\beta \) and hence a sum of roots less than or equal to \(\alpha \).
Therefore without loss of generality we may assume that \(\lambda =m\beta \) and that \(L_\lambda \) is semicuspidal. For want of a contradiction, assume \(\beta \succ \alpha \). We may further assume without loss of generality that \(\beta \) is the maximal root for which \({\text {Res}}_{m\beta ,\nu -m\beta }L\ne 0\) for some positive integer m. We may further assume that m is as large as possible.
By inductive hypothesis, write \(L_\mu =A(M_1,\ldots ,M_n)\) where \(M_1\) is a \(R(k\gamma )\)-module for some root \(\gamma \) and positive integer k.
Therefore \({\text {Res}}_{\lambda +k\gamma ,\mu -k\gamma }L\ne 0\). If \(k\gamma \ne \mu \), then by maximality of \(\beta \), \(\lambda +k\gamma \) is a sum of roots less than or equal to \(\beta \). By maximality of m, \(\gamma \prec \beta \). By adjunction this implies that L is a quotient of \(L_\lambda \circ M_1\circ \cdots \circ M_n\). As \((L_\lambda ,M_1,\ldots ,M_n)\) is an admissible sequence of semicuspidal modules, this is a contradiction.
Therefore \(L_\mu \) is semicuspidal, with \(\mu =k\gamma \). By convexity \(\gamma \prec \alpha \prec \beta \). By adjunction there is a nonzero map from \(L_\lambda \circ L_\mu \) to L. As L is irreducible, this exhibits L as \(A(L_\lambda ,L_\mu )\), a contradiction. This completes the proof. \(\square \)
9 Real cuspidals
For \(i\in I\), there is an automorphism \(T_i\) of the entire quantum group \(U_q({\mathfrak {g}})\) satisfying
for all \(i\ne j\). In the notation of [19], \(T_i\) is the automorphism \(T'_{i,+}\).
Now we will define the PBW root vectors for the real roots. Let \(\alpha \) be a positive real root and suppose that \(\alpha \prec \delta \). Let \(S_\alpha =\{\beta \in \Phi ^+\mid \alpha -\beta \in {\mathbb {N}}I \}\). Then \(S_\alpha \) is a finite set of roots. By Theorem 3.11, we can find a word convex order \(\prec '\) whose restriction to \(S_\alpha \) agrees with the restriction of \(\prec \) to \(S_\alpha \).
By Lemma 3.8 there exists \(w\in W\) such that \(\Phi (w)=\{\beta \in \Phi ^+\mid \beta \preceq '\alpha \}\) and a reduced expression \(w=s_{i_1}\ldots s_{i_N}\) such that \(\alpha =s_{i_1}s_{i_2}\cdots s_{i_{N-1}} \alpha _{i_N}\). We define the root vector \(E_\alpha \in {\mathbf {f}}\) by
If \(\alpha \) happens to be greater than \(\delta \), then in a similar vein we get a reduced expression but now define \(E_\alpha \in {\mathbf {f}}\) by
In all cases, we then define the dual root vector \(E_\alpha ^*=(1-q_\alpha ^2)E_\alpha \in {\mathbf {f}}^*\).
A proof that the elements \(E_\alpha \) and \(E_\alpha ^*\) are well defined based on [19, Proposition 40.2.1] is possible. Alternatively, this result will follow from Theorem 9.1.
For \(\alpha \in \Phi ^+_{{re}}\), let \(L(\alpha )\) be the unique self-dual cuspidal irreducible representation of \(R(\alpha )\). The existence of a cuspidal irreducible module is Theorem 8.9 above while the fact that it can be chosen to be self-dual is in [11, §3.2].
Theorem 9.1
Let \(\alpha \) be a positive real root. Then \([L(\alpha )]=E_\alpha ^*\).
Proof
Let \(i_1,\ldots ,i_N\) be as in the construction of \(E_\alpha \) above. For \(1\le k\le N\) let \(\alpha _k=s_{i_1}\cdots s_{i_{k-1}} \alpha _{i_k}\). Then \(\alpha _1\prec \cdots \prec \alpha _N=\alpha \).
First we will prove by induction on n for \(1\le n\le N\) that there exists \(x_n\in {\mathbf {f}}^*_{{\mathbb {Q}}(q)}\) such that \([L(\alpha )]=T_{i_1}\cdots T_{i_{n-1}}(x_{n})\).
For the case \(n=1\), let \(x_1=[L(\alpha )]\). Now assume that the result is known for \(n=k\) and consider the case \(n=k+1\).
By [19, Ch 38] we can write \(x_k=\theta _{i_k}^* y + T_{i_k}(z)\) where \(y,z\in {\mathbf {f}}^*_{{\mathbb {Q}}(q)}\). Then
Since \(L(\alpha )\) is cuspidal and \(\alpha _k\prec \alpha \), \({\text {Res}}_{\alpha _k,\alpha _N-\alpha _k}L(\alpha )=0\). Therefore \([L(\alpha )]\) is orthogonal to the product \((T_{i_1}\cdots T_{i_{k-1}}(\theta _{i_k}^*)) (T_{i_1}\cdots T_{i_{k-1}}(y))\). Since this product is orthogonal to \(T_{i_1}\cdots T_{i_k}(z)\), it must be that \((y,y)=0\).
If \({\mathbb {Q}}(q)\) is embedded into \({\mathbb {R}}\) by sending q to a sufficiently small real number, then the form \((\cdot ,\cdot )\) on \({\mathbf {f}}_\nu \) is positive definite. Therefore \(y=0\). We let \(x_{k+1}=z\).
We have now proved the desired preliminary result by the principle of mathematical induction. Applying this when \(k=N\), we see that \([L(\alpha )]=T_{i_1}\cdots T_{i_{n-1}}(x_{N})\) for some \(x_N\in ({\mathbf {f}}^*_{{\mathbb {Q}}(q)})_{\alpha _{i_N}}\). Therefore \(x_N\) is a scalar multiple of \(\theta _{i_N}\).
Since \([L(\alpha )]\) is an element of a \({\mathbb {Z}}[q,q^{-1}]\)-basis of \({\mathbf {f}}^*\), the scalar must be a unit in \({\mathbb {Z}}[q,q^{-1}]\), thus of the form \(\pm q^i\) for some \(i\in {\mathbb {Z}}\). Since \([L(\alpha )]\) is invariant under the bar involution, \(i=0\). The argument of [13, Lemma 2.35] shows that the sign is the positive one. \(\square \)
Proposition 9.2
Let \(\alpha \) be a real root and n be a positive integer. The module \(L(\alpha )^{\circ n}\) is the unique simple semicuspidal representation of \(R(n\delta )\).
Proof
By Lemma 8.2, \(L(\alpha )^{\circ n}\) is semicuspidal. Therefore \([L(\alpha )^{\circ n}]=f(q)[L]\) where L is the unique semicuspidal representation of \(R(n\alpha )\) and \(f(q)\in {\mathbb {N}}[q,q^{-1}]\). By Theorem 9.1, \([L(\alpha )^{\circ n}]=T_{i_1}T_{i_2}\cdots T_{i_{N-1}} (\theta _{i_N}^*)^n\) which is indivisible in \({\mathbf {f}}^*\), hence \(L(\alpha )^{\circ n}\) is irreducible. \(\square \)
Remark 9.3
This gives the existence of many modules called real in the nomenclature of [10].
10 Root partitions
Let S be an indexing set for the set of self-dual irreducible semicuspidal representations of \(R(n\delta )\), for all n. It will not be until Theorem 19.10 that we exhibit a bijection between S and \({\mathcal {P}}^\Omega \). We write L(s) for the representation indexed by \(s\in S\).
We now introduce the notion of a root partition, which allows us to index irreducibles by a finite collection of real roots (with multiplicities), together with an irreducible semicuspidal imaginary module. We first define a root partition \(\pi \) to be an admissible sequence of self-dual irreducible semicuspidal representations.
To each root partition \(\pi \) we define a function \(f_\pi \,{:}\,\Phi ^+_{nd}\!\rightarrow \!{\mathbb {N}}\) where if \(f_\pi (\alpha )\) is nonzero then there is a representation of \(R(f_\pi (\alpha )\alpha )\) in \(\pi \). Given two root partitions \(\pi \) and \(\sigma \) we say that \(\pi <\sigma \) if there exist indivisible roots \(\alpha \) and \(\alpha '\) such that \(f_\pi (\alpha )<f_\sigma (\alpha )\), \(f_\pi (\alpha ')<f_\sigma (\alpha ')\) and \(f_\pi (\beta )=f_\sigma (\beta )\) for all roots \(\beta \) satisfying either \(\beta \prec \alpha \) or \(\beta \succ \alpha '\). If \(f_\pi =f_\sigma \) we say \(\pi \sim \sigma \).
Since there is exactly one irreducible semicuspidal representation of \(R(n\alpha )\) for each n and each real root \(\alpha \), we can write the datum of a root partition in a more combinatorial manner. Concretely we write a root partition in the form \(\pi =(\beta _1^{m_1},\ldots ,\beta _k^{m_k},s,\gamma _l^{n_l},\ldots ,\gamma _1^{n_1})\). Here k and l are natural numbers, \(s\in S\), \(\beta _1,\ldots ,\beta _k,\gamma _1,\ldots ,\gamma _l\) are the set of real roots on which \(f_\pi \) is nonzero, \(f_\pi (\beta _i)=m_i\), \(f_\pi (\gamma _i)=n_i\) and
When we do have a bijection between S and \({\mathcal {P}}^\Omega \) then we will have a purely combinatorial description of a root partition.
Let \(\pi =(\beta _1^{m_1},\ldots ,\beta _k^{m_k},s,\gamma _l^{n_l},\ldots ,\gamma _1^{n_1})\) be a root partition. Let \(s_\lambda =\sum _{i=1}^k{\left( {\begin{array}{c}m_i\\ 2\end{array}}\right) }+\sum _{j=1}^l\left( {\begin{array}{c}n_j\\ 2\end{array}}\right) \). Define the proper standard module \(\overline{\Delta }(\pi )\) to be
Let \(L(\pi )\) be the head of \(\overline{\Delta }(\pi )\). This is an irreducible module by Lemma 8.7.
Theorem 10.1
[13] The proper standard modules have the following property.
-
(1)
Up to isomorphism and grading shift, the set \(\{L(\pi )\}\) as \(\pi \) runs over all root partitions of \(\nu \) is a complete and irredundant set of irreducible \(R(\nu )\)-modules.
-
(2)
The module \(L(\pi )\) is self dual, i.e. \(L(\pi )^\circledast \cong L(\pi )\).
-
(3)
If the multiplicity \([\overline{\Delta }(\pi ):L(\sigma )]\) is nonzero, then \(\sigma \le \pi \). Furthermore \([\overline{\Delta }(\pi ):L(\pi )]=1\).
Proof
Part (1) is Theorem 8.8. For part (2) note that since \(L(\pi )\) is irreducible, by [11, §3.2] \(L(\pi )^\circledast \cong q^i L(\pi )\) for some i. By Lemma 8.7(2) and the fact that restriction commutes with duality, \(i=0\). Part (3) follows from Lemma 8.6. \(\square \)
11 Levendorskii–Soibelman formula
By Theorem 10.1 the classes \([\overline{\Delta }(\pi )]\) of the proper standard modules is a basis of \({\mathbf {f}}^*\). We call this the categorical dual PBW basis. Let \(\{E_\pi \}\) be the basis of \({\mathbf {f}}\) dual to this with respect to \(\langle \cdot ,\cdot \rangle \). We shall call this basis the categorical PBW basis. Later we will identify the categorical PBW basis both with a basis coming from a family of standard modules, as well as an algebraically defined basis which generalises the approach of [3].
The results in this section are an affine type analogue of the Levendorskii-Soibelman formula [17, Proposition 5.5.2]. We refer to both Theorems 11.1 and 11.6 as a Levendorskii-Soibelman formula.
Theorem 11.1
Let \(\theta ,\psi \in \Phi ^+_{{re}} \cup S\) with \(\theta \succ \psi \). Expand \([L(\theta )][L(\psi )]-q^{(\theta \cdot \psi )}[L(\psi )][L(\theta )]\) in the standard basis
If \(c_\pi \ne 0\) for some root partition \(\pi \) then \(\pi < (\theta ,\psi )\) where < is the partial order on root partitions from Sect. 10.
Proof
By Theorem 10.1,
Applying the bar involution on \({\mathbf {f}}^*\) yields
Theorem 10.1 also shows that
so upon subtraction we obtain the desired result. \(\square \)
Lemma 11.2
Let \(\sigma \) and \(\pi \) be two root partitions. Then \(\langle [\overline{\Delta }(\sigma )],[\overline{\Delta }(\pi )]\rangle = 0\) unless \(\sigma \sim \pi \).
Proof
We have \(\langle [\overline{\Delta }(\sigma )],[\overline{\Delta }(\pi )]\rangle = \langle [L(\sigma _1)\otimes \cdots \otimes L(\sigma _l)],[{\text {Res}}_\sigma \overline{\Delta }(\pi )]\rangle \) which, by Lemma 8.6, is zero unless \(\sigma \le \pi \).
Also \(\langle [\overline{\Delta }(\sigma )],[\overline{\Delta }(\pi )]\rangle =\langle [{\text {Res}}_\pi (\overline{\Delta }(\sigma )],[L(\pi _1)\circ \cdots \circ L(\pi _k)]\rangle \) which again by Lemma 8.6 is zero unless \(\pi \le \sigma \). \(\square \)
Lemma 11.3
If \(\theta \in \Phi ^+_{re}\cup S\) then
Proof
By definition of \(E_\theta \), we have \(\langle E_\theta ,[\overline{\Delta }(\pi )]\rangle = 0\) unless \(\pi =\theta \). By Lemma 11.2 and the fact that the classes \([\overline{\Delta }(\sigma )]\) are a basis of \({\mathbf {f}}^*\), this forces
\(\square \)
Corollary 11.4
If \(\sigma ,\pi \in S\) then \(E_\sigma E_\pi \) is a linear combination of \(E_\tau \) for \(\tau \in S\).
Proof
This follows from Lemmas 8.2 and 11.3. \(\square \)
Lemma 11.5
Let \(\pi =(\pi _1,\ldots ,\pi _k)\) be a root partition. Then \(E_\pi =E_{\pi _1}\cdots E_{\pi _k}\).
Proof
By Lemma 11.3, the element \(E_{\pi _1}\cdots E_{\pi _k}\) is a linear combination of elements of the form \([\overline{\Delta }(\sigma )]\) where \(\sigma \sim \pi \). Therefore by Lemma 11.2, \(E_{\pi _1}\cdots E_{\pi _k}\) is orthogonal to all elements of the from \([\overline{\Delta }(\eta )]\) where \(\eta \not \sim \pi \). For \(\eta \sim \pi \), we compute
which is zero unless \(\eta =\pi \) in which case it is equal to one. We have shown that the product \(E_{\pi _1}\cdots E_{\pi _k}\) has all the properties which define \(E_\pi \), hence is equal to \(E_\pi \). \(\square \)
Theorem 11.6
Let \(\theta ,\psi \in \Phi ^+_{{re}} \cup S\) with \(\theta \succ \psi \). Then
Proof
This is immediate from Theorem 11.1 and Lemma 11.3. \(\square \)
This yields an algorithm for expanding any monomial in the \(E_\theta \) in the PBW basis. Namely given a monomial \(E_{\kappa _1}E_{\kappa _2}\cdots E_{\kappa _k}\), repeatedly apply the following types of moves:
-
If \(\kappa _l \prec \kappa _{l+1}\), replace \(E_{\kappa _l}E_{\kappa _{l+1}}\) with \(q^{\kappa _l\cdot \kappa _{l+1}}E_{\kappa _{l+1}}E_{\kappa _{l}}\) plus the correction term from Theorem 11.6.
-
If \(\kappa _l\) and \(\kappa _{l+1}\) are both imaginary replace the product \(E_{\kappa _l}E_{\kappa _{l+1}}\) with a sum of terms \(E_\theta \), which is possible by Corollary 11.4.
12 Minimal pairs
Let \(\alpha \) be a positive root. Define \(S(\alpha )\) to be the quotient of \(R(\alpha )\) by the two-sided ideal generated by the set of \(e_\mathbf{i}\) such that \(e_\mathbf{i}L=0\) for all semicuspidal modules L.
Lemma 12.1
There is an equivalence of categories between the category of \(S(\alpha )\) modules and the full subcategory of semicuspidal \(R(\alpha )\)-modules.
Proof
It is clear from the definition that any semicuspidal \(R(\alpha )\)-module is a \(S(\alpha )\)-module.
Conversely suppose that M is a \(S(\alpha )\)-module. Suppose that \(\lambda =(\lambda _1,\ldots ,\lambda _l)\) is a root partition such that \(q^n L(\lambda )\) appears as a subquotient of M. Then \(e_\mathbf{i}M\ne 0\) for some \(\mathbf{i}\) which is the concatenation of \(\mathbf{i}_1,\ldots ,\mathbf{i}_l\) in \(\,\hbox {Seq}\,(\lambda _1),\ldots ,\,\hbox {Seq}\,(\lambda _l)\) respectively. If \(\lambda \ne \alpha \), then \(e_\mathbf{i}L=0\) for all semicuspidal \(R(\alpha )\)-modules L. Therefore \(e_\mathbf{i}\) has zero image in \(S(\alpha )\), contradicting \(e_\mathbf{i}M\ne 0\). Hence all composition factors of M are semicuspidal, so M is semicuspidal. \(\square \)
Definition 12.2
Let \(\alpha \) be a positive root. A minimal pair for \(\alpha \) is an ordered pair of roots \((\beta ,\gamma )\) satisfying \(\alpha =\beta +\gamma \), \(\gamma \prec \beta \) and there is no pair of roots \((\beta ',\gamma ')\) satisfying \(\alpha =\beta '+\gamma '\) and \(\gamma \prec \gamma '\prec \beta '\prec \beta \).
Lemma 12.3
Let \(\alpha \) be a positive root and let \((\beta ,\gamma )\) be a minimal pair for \(\alpha \). Let L be a cuspidal representation of \(R(\alpha )\). Then \({\text {Res}}_{\gamma ,\beta }L\) is a \(S(\gamma )\otimes S(\beta )\)-module.
Proof
Expand \([{\text {Res}}_{\gamma ,\beta }L]\) in the categorical dual PBW basis
Then
In the previous section we showed how the Levendorskii-Soibelman formula gave an algorithm for expanding the product \(E_\pi E_\sigma \) into the PBW basis. Each term \(E_{\kappa _1}\cdots E_{\kappa _n}\) which appears at some point in this expansion has \(\kappa _1\succeq \pi _1\succeq \beta \) and \(\kappa _n\preceq \sigma _l\preceq \gamma \).
The only PBW basis elements which fail to be orthogonal to [L] are those of the form \(E_\alpha \) if \(\alpha \) is real and \(E_s\) with \(s\in S\) if \(\alpha \) is imaginary. For such a term to appear, it must arise as a result of applying Theorem 11.6 to a term \(E_{\kappa _1}E_{\kappa _2}\) with \(\kappa _1+\kappa _2=\alpha \).
We have already showed that \(\kappa _1\succeq \beta \) and \(\kappa _2\preceq \gamma \). To apply the Levendorskii-Soibelman formula we need \(\kappa _1\prec \kappa _2\) and we also know \(\kappa _1+\kappa _2=\alpha \). Since \((\beta ,\gamma )\) is a minimal pair, this forces \(\kappa _1=\beta \) and \(\kappa _2=\gamma \). Therefore the coefficient \(c_{\pi \sigma }\) can only be nonzero if \(\pi =\kappa _1\) and \(\sigma =\kappa _2\). Hence \([{\text {Res}}_{\gamma \beta }L]\) is a linear combination of elements of the form \([L_\gamma ]\otimes [L_\beta ]\) where \(L_\gamma \) and \(L_\beta \) are cuspidal representations of \(R(\gamma )\) and \(R(\beta )\). This implies that \({\text {Res}}_{\gamma \beta }L\) is a \(S(\gamma )\otimes S(\beta )\)-module, as required. \(\square \)
A chamber coweight \(\omega \) is said to be adapted to the convex order \(\prec \) if it is a fundamental coweight for the positive system \(p(\Phi _{\succ \delta })\) in \(\Phi _f\). Let \(\omega \) be such a chamber coweight. Then there exists a root \(\alpha \in p(\Phi _{\succ \delta })\) such that \(\langle \omega ,\alpha \rangle =1\) and \(\langle \omega ,\beta \rangle =0\) for all \(\beta \in p(\Phi _{\succ \delta })\setminus \{\alpha \}\). Let \(\omega _+=\tilde{\alpha }\) and \(\omega _-=\widetilde{-\alpha }\). We will always assume that all chamber coweights are adapted to the given convex order.
Lemma 12.4
Let \(\alpha \) be a positive real root which is not simple that does not have a real minimal pair. Then there exists a chamber coweight \(\omega \) adapted to \(\prec \) such that \(\alpha =\omega _++n\delta \) or \(\alpha =\omega _-+n\delta \) for some \(n\in {\mathbb {N}}\).
Proof
Since every root which is not simple has a minimal pair, if \(\alpha \) has no real minimal pair it must be that \(\alpha -\delta \) is also a root.
Without loss of generality suppose \(\alpha \succ \delta \). The \(p(\alpha )\) is a positive root in \(\Phi _f\). We have to prove that \(p(\alpha )\) is simple. Suppose for want of a contradiction that \(p(\alpha )=\beta +\gamma \) for two positive roots \(\beta \) and \(\gamma \). Then \(\alpha =\tilde{\beta }+\tilde{\gamma }+n\delta \) for some n. If \(n\ge 0\) then we can use this expression to write \(\alpha \) as a sum of two roots both greater than \(\delta \), which proves that \(\alpha \) has a real minimal pair.
Therefore the only case left to consider is if \(\tilde{\beta }+\tilde{\gamma }-2\delta \) is a positive root. When writing \(\tilde{\beta }+\tilde{\gamma }\) in the form \(n\delta +x\) with \(x\in \Phi _f\), \(n\ge 2\) with equality if and only if \(\beta \) and \(\gamma \) are negative under the usual positive system on \(\Phi _f\). Therefore it is impossible for \(\tilde{\beta }+\tilde{\gamma }-2\delta \) to be a root. \(\square \)
13 Independence of convex order
In this section, we prove some results detailing how some modules which a priori depend on the entire convex order \(\prec \), only depend on the positive system \(p(\Phi _{\prec \delta })\).
Theorem 13.1
Let \(\omega \) be a chamber coweight. The algebras \(S(\omega _+)\) and \(S(\omega _-)\) only depend on the set \(p(\Phi _{\prec \delta })\).
Remark 13.2
For a balanced convex order, this is [13, Lemma 5.2].
Proof
It suffices to prove that the simple modules \(L(\omega _-)\) and \(L(\omega _+)\) depend only on \(p(\Phi _{\prec \delta })\).
We write \(E_\alpha ^\prec \) for the root vector defined using the convex order \(\prec \). Let \(\prec \) and \(\prec '\) be two convex orders with \(p(\Phi _{\prec \delta })=p(\Phi _{\prec ' \delta })\). Without loss of generality we may assume that \(\prec \) and \(\prec '\) are of word type. Label the roots smaller than \(\delta \) as \(\alpha _1\prec \alpha _2\prec \cdots \) and \(\alpha _1'\prec '\alpha _2'\prec '\cdots \). Let n and N be such that
Let w be the element of W such that \(\Phi (w)=\{\alpha _1,\ldots ,\alpha _n\}\) and let \(u\in W\) be such that \(\Phi (u)=\{\alpha _1',\ldots ,\alpha _N'\}\). Then \(\Phi (w)\subset \Phi (u)\). Hence if we fix a reduced expression for w (in particular the one used to define \(E_{\omega _-}^\prec \)) then there exists a reduced expression for u beginning with this fixed reduced expression for w.
By [19, Prop 40.2.1] there exists a subspace \(U^+(u)\) of \({\mathbf {f}}\) which contains \(E_{\omega _-}^\prec \) and \(E_{\omega _-}^{\prec '}\). The dimension of \(U^+(u)_{\omega _-}\) is equal to the number of ways of writing \(\omega _-\) as a \({\mathbb {N}}\)-linear combination of roots in \(\Phi ^+(u)\). Any nontrivial expression contradicts the simplicity of \(p(\omega _-)\), hence this space is one-dimensional, so \(E_{\omega _-}^\prec \) and \(E_{\omega _-}^{\prec '}\) are scalar multiples of one another.
By Theorem 9.1, \((1-q^2)E_{\omega _-}\) is the character of the irreducible module \(L(\omega _-)\), hence this scalar must be one and the module \(L(\omega _-)\) is the same for the convex orders \(\prec \) and \(\prec '\). This completes the proof for \(\omega _-\) and the proof for \(\omega _+\) is similar. \(\square \)
Lemma 13.3
Let \((\beta ,\gamma )\) be a minimal pair for \(\delta \). Let \(L_\beta \) and \(L_\gamma \) be cuspidal \(R(\beta )\) and \(R(\gamma )\)-modules respectively. Then \({\text {Res}}_{\gamma \beta }(L_\gamma \circ L_\beta ) \cong L_\gamma \otimes L_\beta \) and \({\text {Res}}_{\gamma \beta }(L_\beta \circ L_\gamma ) \cong q^{-\beta \cdot \gamma } L_\gamma \otimes L_\beta \).
Proof
By Lemma 13.1, without loss of generality, assume our convex order \(\prec \) is as in Example 3.6. Thus the only roots between \(\gamma \) and \(\beta \) are of the form \(\gamma +n\delta \), \(\beta +n\delta \) or \(n\delta \).
Consider a nonzero quotient in the Mackey filtration of \({\text {Res}}_{\gamma ,\beta }(L_\gamma \circ L_\beta )\). Then we have \(\lambda ,\mu ,\nu \in {\mathbb {N}}I\) such that \(\lambda +\mu =\gamma \), \(\mu +\nu =\beta \), \(\lambda \) is a sum of roots less than or equal to \(\gamma \), \(\nu \) is a sum of roots greater than or equal to \(\beta \), while \(\mu \) is both a sum of roots greater than or equal to \(\gamma \) and a (possibly different) sum of roots less than or equal to \(\beta \).
Consider \(\gamma =\lambda +\mu \) which has been written as a sum of roots less than or equal to \(\beta \). No roots between \(\gamma \) and \(\beta \) can appear in this sum. By convexity of the convex order, the only options are \(\mu =0\), \(\mu =\gamma \) and \(\mu =\beta \). We will have to show that the last two options are not possible.
So suppose for want of a contradiction that \(\mu =\gamma \). Then \(\nu =\beta -\gamma =\sum _i \nu _i\) with each \(\nu _i\) larger than \(\beta \). Note that there is at least two terms in this sum as \(\beta -\gamma \) is not a root.
Since \((\gamma ,\beta -\gamma )=-4\), there exists an index j such that \((\gamma ,\nu _j)<0\). Therefore \(\gamma +\nu _j\) is a root. Now consider
By convexity this implies \(\gamma +\nu _j\prec \beta \) and as \(\nu _j\succ \beta \succ \gamma \) it must be that \(\gamma +\nu _j\succ \gamma \). The Eq. (13.1) implies \(|\gamma +\nu _j|<|\beta |\). But on the other hand we’ve classified all roots \(\alpha \) between \(\beta \) and \(\gamma \) and none of them satisfy \(|\alpha |<\beta \), a contradiction. The case \(\mu =\beta \) is handled similarly.
Therefore there is only one term in the Mackey filtration, which is the one where \(\mu =0\), whence we obtain the lemma. \(\square \)
14 Simple imaginary modules
We start by following [9] and defining the R-matrices for KLR algebras. First we need to introduce some useful elements of \(R(\nu )\).
For \(1\le a< n=|\nu |\) we define elements \(\varphi _a\in R(\nu )\) by
These elements satisfy the following properties
Lemma 14.1
[9, Lemma 1.3.1]
-
(1)
\(\varphi _a^2e_\mathbf{i}=( Q_{\nu _a,\nu _{a+1}}(x_a,x_{a+1})+\delta _{\nu _a,\nu _{a+1}}) e_\mathbf{i}.\)
-
(2)
\(\{\varphi _k\}_{1\le k<n}\) satisfies the braid relations.
-
(3)
For \(w\in S_n\), let \(w=s_{a_1}\cdots s_{a_\ell }\) be a reduced expression of w and set \(\varphi _w=\varphi _{a_1}\cdots \varphi _{a_\ell }\). Then \(\varphi _w\) does not depend on the choice of reduced expressions of w.
-
(4)
For \(w\in S_n\) and \(1\le k\le n\), we have \(\varphi _w x_k=x_{w(k)}\varphi _w\).
-
(5)
For \(w\in S_n\) and \(1\le k<n\), if \(w(k+1)=w(k)+1\), then \(\varphi _w\tau _k=\tau _{w(k)}\varphi _w\).
-
(6)
\(\varphi _{w^{-1}}\varphi _we_\mathbf{i}=\prod \nolimits _{ \begin{array}{c} a<b,\\ w(a)>w(b) \end{array} } (Q_{\mathbf{i}_a,\mathbf{i}_b}(x_a,x_b)+\delta _{\mathbf{i}_a,\mathbf{i}_b})e_\mathbf{i}\).
Let M and N be modules for \(R(\lambda )\) and \(R(\mu )\) respectively. Let \((\lambda ,\mu )_n\) be the degree of \(\phi _{w[\lambda ,\mu ]}\). Define the morphism \(R_{M,N}\,{:}\,M\circ N\!\rightarrow \!q^{-(\lambda ,\mu )_n}N\circ M\) by
In [9] an algebra homomorphism \(\psi _z\,{:}\,R(\nu )\!\rightarrow \!{\mathbb {Q}}[z]\otimes R(\nu )\) is constructed where \(\psi _z(e_\mathbf{i})=e_\mathbf{i}\), \(\psi _z(y_j)=y_j+z\) and \(\psi _z(\tau _k)=\tau _k\). If M is an \(R(\nu )\)-module we define the \(R(\nu )\)-module \(M_z=\psi _z^*({\mathbb {Q}}[z]\otimes M)\). The morphism \(r_{M,N}\,{:}\,M\circ N\!\rightarrow \!q^{2s-(\lambda ,\mu )_n}N\circ M\) is now defined by
where s is the largest possible integer for which this definition is possible. In [9] it is shown that \(r_{M,N}\) is a nonzero morphism and that these collections of morphisms satisfy the braid relation.
Lemma 14.2
Let \(L_1\) and \(L_2\) be two irreducible cuspidal representations of \(R(\delta )\). Then the morphisms \(r_{L_1,L_2}\) and \(r_{L_2,L_1}\) are inverse to one another.
Proof
By adjunction
As \(L_1\) and \(L_2\) are cuspidal, the Mackey filtration of \({\text {Res}}_{\delta ,\delta } (L_2\circ L_1)\) has two nonzero pieces, namely \(L_2\otimes L_1\) and \(L_1\otimes L_2\). In particular this implies that \({\text {Hom}}(L_1\circ L_2,L_2\circ L_1)\) is concentrated in degree zero. Since \(r_{L_1,L_2}\ne 0\), the integer s in the construction of \(r_{L_1,L_2}\) must be equal to \((\delta ,\delta )_n/2\).
For \(j=1,2\), pick a nonzero vector \(v_j\in L_j\) such that \(y_i v_j=0\) for all i. The morphism \(r_{L_2,L_1}r_{L_1,L_2}\) maps \(v_1\otimes v_2\) to \(\left( (z'-z)^{-2s}\varphi _{w[\delta ,\delta ]}^2 v_1\otimes v_2\right) |_{z=z'=0}\) where the computation is taking place in \((L_1)_{z}\circ (L_2)_{z'}\) (by abuse of notation, we write v for \(1\otimes v\in L_z\)). We can compute this using Lemma 14.1(vi). Since \(y_i v_j=0\) in \(L_j\), we have \(y_i v_j=zv_j\) in \((L_j)z\). Then the product on the right hand side of 14.1(vi) acts by the scalar \((z'-z)^{(\delta ,\delta )_n}\) on the vector \(v_1\otimes v_1\in (L_1)_{z}\circ (L_2)_{z'}\). We’ve already computed \((\delta ,\delta )_n=2s\) and hence \(r_{L_2,L_1}r_{L_1,L_2} v_1\otimes v_2=v_1\otimes v_2\).
Since \(L_1\) and \(L_2\) are irreducible, \(v_1\otimes v_2\) generates \(L_1\circ L_2\). Therefore \(r_{L_2,L_1}r_{L_1,L_2}\) is the identity. \(\square \)
From the evident maps from \({\text {End}}(L\circ L)\) to \({\text {End}}(L^{\circ n})\), the morphisms \(r_{L,L}\) define \(n-1\) elements, denoted \(r_1,r_2,\ldots , r_{n-1} \in {\text {End}}(L^{\circ n})\). The following result was first noticed in a special case in [15, Theorem 4.13], and is fundamental to the paper [14].
Theorem 14.3
Let L be an irreducible cuspidal representation of \(R(\delta )\). There is an isomorphism \({\text {End}}(L^{\circ {n}})\cong {\mathbb {Q}}[S_n]\) sending \(r_i\) to the transposition \((i,i+1)\).
Proof
By adjunction \({\text {End}}(L^{\circ n})={\text {Hom}}(L^{\otimes n},{\text {Res}}_{\delta ,\ldots ,\delta }L^{\circ n})\). Since L is cuspidal, the Mackey filtration of \({\text {Res}}_{\delta ,\ldots ,\delta }L^{\circ n}\) has exactly n! nonzero subquotients, each isomorphic to \(L^{\otimes n}\). Therefore \(\dim {\text {End}}(L^{\circ n})\le n!\).
By Lemma 14.2, \(r_i^2=1\). The identity \(r_ir_j=r_jr_i\) for \(|j-i|>1\) is trivial and the braid relation \(r_i r_{i+1} r_i= r_{i+1} r_i r_{i+1}\) is a general fact about the morphisms \(r_{M,N}\) constructed in [9]. This allows us to define \(r_w\) for each \(w\in S_n\).
Recall that in the proof of Lemma 14.2, we showed that \(s=(\delta ,\delta )_n/2\), where s is the integer appearing in the definition of \(r_{L,L}\). Therefore by induction on the length of w, using [9, Proposition 1.4.4(iii)], we obtain
where \(\iota \,{:}\,S_n\!\rightarrow \!S_{n|\delta |}\) is the obvious embedding. Therefore the endomorphisms \(r_w\) are linearly independent.
Since the \(r_i\) satisfy the Coxeter relations there is a homomorphism from \({\mathbb {Q}}[S_n]\) to \({\text {End}}(L^{\circ n})\). We have just shown it is injective. Surjectivity follows from the dimension estimate in the first paragraph of this proof. \(\square \)
Let \(\omega \) be a chamber coweight. Let \(L(\omega )\) be the head of the module \(L(\omega _-)\circ L(\omega _+)\).
Lemma 14.4
The module \(L(\omega )\) is an irreducible module with \(L(\omega )^\circledast \cong L(\omega )\). Furthermore \({\text {Res}}_{\omega _-,\omega _+}L(\omega )\cong L(\omega _-)\otimes L(\omega _+)\).
Remark 14.5
The irreducibility of \(L(\omega )\) is in [25] and can also be derived from [10, Theorem 3.2]. Our preference for giving this proof is that we wish to make use of the extra properties of \(L(\omega )\) that we establish.
Proof
Using Theorem 13.1 and the convex order from Example 3.6, we may assume without loss of generality that \((\omega _+,\omega _-)\) is a minimal pair for \(\delta \).
For any quotient Q of \(L(\omega _-)\circ L(\omega _+)\) there is, by adjunction, a nonzero morphism from \(L(\omega _-)\otimes L(\omega _+)\) to \({\text {Res}}_{\omega _-,\omega _+}Q\) which is injective as the source is irreducible. Lemma 13.3 implies that \({\text {Res}}_{\omega _-,\omega _+} L(\omega _-)\circ L(\omega _+)\cong L(\omega _-)\otimes L(\omega _+)\). By exactness of the restriction functor, this forces the head of \(L(\omega _-)\circ L(\omega _+)\) to be irreducible and furthermore \({\text {Res}}_{\omega _-,\omega _+}L(\omega )\cong L(\omega _-)\otimes L(\omega _+)\). The self-duality of \(L(\omega )\) follows since every simple module is self-dual up to a grading shift, duality commutes with restriction and the modules \(L(\omega _\pm )\) are self-dual. \(\square \)
15 The growth of a quotient
Let z be the element \(y_1+\cdots +y_{|\nu |}\in R(\nu )\). It is straightforward to check that z is central. The following lemma and proof appeared in an early version of [4].
Lemma 15.1
Let \(R'(\nu )\) be the subalgebra of \(R(\nu )\) generated by \(e_\mathbf{i}\), \(\mathbf{i}\in \,\hbox {Seq}\,(\nu )\), \(\tau _i\) and \(y_i-y_{i+1}\), \(1\le i<|\nu |\). Then multiplication induces an algebra isomorphism \({\mathbb {Q}}[z]\otimes R'(\nu )\rightarrow R(\nu )\).
Proof
An inspection of the presentation (5.1) of \(R(\nu )\) shows that the set of elements of the form
with \(a_1,\ldots ,a_{n-1}\in {\mathbb {N}}\), \(w\in S_n\) and \(\mathbf{i}\in \,\hbox {Seq}\,(\nu )\) is a spanning set for \(R'(\nu )\). Since Theorem 5.4 provides us with a basis of \(R(\nu )\), we can see that the collection of elements above forms a linearly independent set, hence is a basis for \(R'(\nu )\). We compute
and thus \(y_n\) is in the image of \({\mathbb {Q}}[z]\otimes R'(\nu )\). Therefore the multiplication map from \({\mathbb {Q}}[z]\otimes R'(\nu )\) to \(R(\nu )\) is surjective. A dimension count using Lemma 5.4 shows that it must be an isomorphism. \(\square \)
Lemma 15.2
Let \(\alpha \) be a positive root. There is an injection from \({\mathbb {Q}}[z]\) into the centre of \(S(\alpha )\).
Proof
Let \(S'(\alpha )\) be the quotient of \(R'(\alpha )\) by the two sided ideal generated by all \(e_\mathbf{i}\) such that \(e_\mathbf{i}L=0\) for all cuspidal representations L of \(R(\alpha )\). Lemma 15.1 implies that \(S(\alpha )\cong {\mathbb {Q}}[z]\otimes S'(\alpha )\). The image of \({\mathbb {Q}}[z]\otimes {\mathbb {Q}}\) provides us with our desired central subalgebra. \(\square \)
Let \(\alpha \) be an indivisible root, L a cuspidal representation of \(R(\alpha )\) and let \((\beta ,\gamma )\) be a minimal pair for \(\alpha \). Let \(L''\otimes L'\) be an irreducible subquotient of \({\text {Res}}_{\gamma ,\beta }L\). By Lemma 12.3, \(L''\) and \(L'\) are cuspidal modules for \(R(\gamma )\) and \(R(\beta )\). We will call \((L',L'')\) a minimal pair for L. We inductively define a word \(\mathbf{i}_L\in \,\hbox {Seq}\,(\alpha )\) as the concatenation \(\mathbf{i}_{L''}\mathbf{i}_{L'}\).
Let T(L) be the subalgebra of \(e_{\mathbf{i}_L} S(\alpha ) e_{\mathbf{i}_L}\) generated by \(y_1 e_{\mathbf{i}_L},\ldots ,y_{|\alpha |}e_{\mathbf{i}_L}\).
Lemma 15.3
Let L be a cuspidal representation and \((L',L'')\) be a minimal pair for L. The inclusion \(R(\gamma )\otimes R(\beta )\rightarrow R(\alpha )\) induces a homomorphism from \(T(L'')\otimes T(L')\) to T(L).
Proof
Suppose \(x\in \ker (R(\gamma )\rightarrow S(\gamma ))\). Consider \(x\otimes 1\in R(\gamma )\otimes R(\beta )\hookrightarrow R(\alpha )\). On \(M\in S(\alpha )\hbox {-mod}\), \(x\otimes 1\) acts in the way it does on \({\text {Res}}_{\gamma \beta }M\), which is a \(S(\gamma )\otimes S(\beta )\)-module. Therefore \(x\otimes 1\) acts by zero and hence is in the kernel of \(R(\alpha )\rightarrow S(\alpha )\). \(\square \)
Lemma 15.4
Let L be a cuspidal representation of \(\alpha \). The scheme \({\text {Proj}}T(L)\) has a unique \(\overline{{\mathbb {Q}}}\)-point \([x_1:\cdots :x_{|\alpha |}]\), namely \(x_1=\cdots =x_{|\alpha |}\).
Proof
We prove this by induction on the height of \(\alpha \). Choose a minimal pair \((\beta ,\gamma )\) for \(\alpha \) and \((L',L'')\) for L. Suppose that \([x_1:\cdots : x_{|\alpha |}]\) is a \(\overline{{\mathbb {Q}}}\)-point of \({\text {Proj}}T(L)\). Then by Lemma 15.3 \([x_1:\cdots : x_{|\gamma |}]\) and \([x_{|\gamma |+1}:\cdots :x_{|\alpha |}]\) are points in \({\text {Proj}}T(L'')\) and \({\text {Proj}}T(L')\) respectively. By inductive assumption, \(x_1=\cdots = x_{|\gamma |}\) and \(x_{|\gamma |+1}=\cdots = x_{|\alpha |}\).
Let \(w=w[|\gamma |,|\beta |]\) and consider the element \(\varphi _w^2 e_{\mathbf{i}_L}\). By Lemma 14.1(vi) it lives in \(T(\mathbf{i}_L)\) and since \(\varphi _w^2 e_{\mathbf{i}_L}=\varphi _w e_{\mathbf{i}_{L'}\mathbf{i}_{L''}}\varphi _w\), it lives in the kernel of the map from \(R(\alpha )\) to \(S(\alpha )\). Therefore \(\varphi _w^2 e_{\mathbf{i}_L}\) is zero in \(T(\mathbf{i}_L)\). Lemma 14.1(vi) writes \(\varphi _w^2 e_{\mathbf{i}_L}\) as a product of elements of the form \(x_i-x_j\) where \(i\le |\gamma |\) and \(j>|\gamma |\). Therefore any \(\overline{{\mathbb {Q}}}\)-point of \({\text {Proj}}T(\mathbf{i}_L)\) has \(x_1=\cdots =x_{|\alpha |}\) as required. \(\square \)
Theorem 15.5
Let \(\alpha \) be an indivisible root. Then \(\dim S(\alpha )_d\) is bounded as a function of d.
Proof
Consider a composition series for \(S(\alpha )\) as a \(S(\alpha )\)-module. Every composition factor must be cuspidal, so
where \(f_L(q)\in {\mathbb {N}}((q))\) and the sum is over irreducible cuspidal representations L. For any \(\mathbf{i}\in Seq(\nu )\), we therefore get the equality
Pick an irreducible cuspidal representation L and let \(\mathbf{i}_L\) be the corresponding word in \(\,\hbox {Seq}\,(\nu )\). By Lemma 15.4 and the theory of the Hilbert polynomial, \(\dim T(\mathbf{i}_L)_d\) is a bounded function of d. From Theorem 5.4 we see that \(e_{\mathbf{i}_L} S(\alpha )\) is finite over \(T(\mathbf{i}_L)\) and hence \(\dim (e_{\mathbf{i}_L} S(\alpha ))_d\) is a bounded function of d.
We take \(\mathbf{i}=\mathbf{i}_L\) in (15.2) and since \(e_{\mathbf{i}_L}L\ne 0\), the Laurent series \(f_L(q)=\sum _d f_L^{(d)} q^d\) has \(f_L^{(d)}\) a bounded function of d. Equation (15.1) completes the proof. \(\square \)
16 An important short exact sequence
Let \(\alpha \) be a real root. Define \(\Delta (\alpha )\) to be the projective cover of \(L(\alpha )\) in the category of \(S(\alpha )\)-modules. Let \(\omega \) be a chamber coweight. Define \(\Delta (\omega )\) to be the projective cover of \(L(\omega )\) in the category of \(S(\delta )\)-modules.
Lemma 16.1
Let \(\alpha \) be an indivisible root. Suppose that \((\beta ,\gamma )\) is a minimal pair for \(\alpha \). Let \(\Delta _\beta \) and \(\Delta _\gamma \) be finitely generated projective \(S(\beta )\) and \(S(\gamma )\)-modules. Then there is a short exact sequence
for some projective \(S(\alpha )\)-module C.
Proof
By adjunction,
Since the modules \(\Delta _\gamma \) and \(\Delta _\beta \) are cuspidal, the Mackey filtration of \({\text {Res}}_{\beta \gamma }\Delta _\gamma \circ \Delta _\beta \) has only one nonzero term, yielding an isomorphism
Let \(\phi \,{:}\,q^{-\beta \cdot \gamma }\Delta _\beta \circ \Delta _\gamma \!\rightarrow \!\Delta _\gamma \circ \Delta _\beta \) be the image of the identity map on \(q^{-\beta \cdot \gamma }\Delta _\beta \otimes \Delta _\gamma \) under the isomorphisms discussed above.
This map \(\phi \) satisfies
for all \(v_\beta \in \Delta _\beta \) and \(v_\gamma \in \Delta _\gamma \).
There are filtrations of \(\Delta _\beta \) and \(\Delta _\gamma \) where each successive subquotient is an irreducible cuspidal module for \(R(\beta )\) or \(R(\gamma )\) respectively. This induces a pair of filtrations on \(\Delta _\beta \otimes \Delta _\gamma \) and \(\Delta _\gamma \otimes \Delta _\beta \) where the successive subquotients are of the form \(L_\beta \circ L_\gamma \) or \(L_\gamma \circ L_\beta \) for cuspidal irreducible representations \(L_\beta \) and \(L_\gamma \) of \(R(\beta )\) and \(R(\gamma )\).
From the explicit formula (16.1), we see that \(\phi \) induces a morphism \(\bar{\phi }\) on each subquotient \(\bar{\phi }\,{:}\,q^{-\beta \cdot \gamma }L_\beta \circ L_\gamma \!\rightarrow \!L_\gamma \circ L_\beta \) satisfying
By Theorem 10.1, the module \(L_\beta \circ L_\gamma \) has an irreducible head \(A(L_\beta ,L_\gamma )\). Since \((\beta ,\gamma )\) is a minimal pair, all other composition factors are cuspidal. Taking duals, \(q^{\beta \cdot \gamma }L_\gamma \circ L_\beta \) has \(A(L_\beta ,L_\gamma )\) as its socle with all other composition factors cuspidal.
The morphism \(\bar{\phi }\) therefore sends the head of \(q^{-\beta \cdot \gamma }L_\beta \circ L_\gamma \) onto the socle of \(L_\gamma \circ L_\beta \). Hence \(\phi \) induces a bijection between all occurrences of non-cuspidal subquotients as sections of filtrations of \(q^{-\beta \cdot \gamma }\Delta _\beta \circ \Delta _\gamma \) and \(\Delta _\gamma \circ \Delta _\beta \). This shows that \(\ker \phi \) and \({\text {coker}}\,\phi \) are both cuspidal \(R(\alpha )\)-modules.
Suppose for want of a contradiction that \(\ker \phi \) is nonzero. It is a submodule of the finitely generated module \(q^{-\beta \cdot \gamma }\Delta _\beta \circ \Delta _\gamma \). By [11, Corollary 2.11], \(R(\alpha )\) is Noetherian and hence \(\ker \phi \) is finitely generated.
As \(\ker \phi \) is cuspidal it is a \(S(\alpha )\)-module, so by Theorem 15.5 we deduce that \(\dim (\ker \phi )_d\) is bounded as a function of d.
The adjunction (6.2) yields a canonical nonzero map from \({\text {Res}}_{\gamma ,\beta }\ker \phi \) to \(\Delta _\gamma \otimes \Delta _\beta \). If X is the image of this map then we have \(\dim X_d\) is a bounded function of d.
The modules \(\Delta _\beta \) and \(\Delta _\gamma \) are free over the central subalgebra \({\mathbb {Q}}[z]\) of \(S(\beta )\) and \(S(\gamma )\). Therefore \(\Delta _\beta \otimes \Delta _\gamma \) is a free \({\mathbb {Q}}[z_1,z_2]\)-module. Hence there are no nonzero submodules M of \(\Delta _\beta \otimes \Delta _\gamma \) for which \(\dim M_d\) is a bounded function of d. This is a contradiction, implying \(\phi \) is injective.
Now let L be a cuspidal \(R(\alpha )\)-module. We apply \({\text {Hom}}(-,L)\) to the short exact sequence
and obtain a long exact sequence. As \({\text {Res}}_{\beta ,\gamma }L=0\), we have
Therefore our long exact sequence degenerates into a sequence of isomorphisms
and by adjunction we have
Lemma 12.3 shows that \({\text {Res}}_{\gamma ,\beta }L\) is a \(S(\gamma )\otimes S(\beta )\)-module. Since \(\Delta _\gamma \otimes \Delta _\beta \) is a projective \(S(\gamma )\otimes S(\beta )\)-module, we derive that \({\text {Ext}}^1(\Delta _\gamma \otimes \Delta _\beta ,{\text {Res}}_{\gamma ,\beta }L)=0\). Tracing through the above isomorphisms yields \({\text {Ext}}^1({\text {coker}}\,\phi ,L)=0\) and therefore \({\text {coker}}\,\phi \) is a projective \(S(\alpha )\)-module. \(\square \)
17 Cuspidal representations of \(R(\delta )\)
We first explain the intertwined logical structure of this section and the following one. Each statement in Sect. 18 involves a positive root \(\alpha \). We prove all the results in this section under an assumption that the results in Sect. 18 are known for all roots \(\alpha \) of height less than the height of \(\delta \). The reader will not be worried about the forward references once the logical structure of Sect. 18 is known.
The results of Sect. 18 will be proved by a simultaneous induction on the height of the root \(\alpha \). In particular, when Theorem 18.1 is proved for a root \(\alpha \), it will be safe to assume that Theorem 18.2 is known for all roots of smaller height. There are references to the results of this section in Sect. 18. However they only appear when the root \(\alpha \) under question is of height at least that of \(\delta \). Thus there is no circularity and the argument is valid.
Let \(\omega \) be a chamber coweight. Recall from Sect. 14 that \(L(\omega )\) is the head of the module \(L(\omega _-)\circ L(\omega _+)\) and is irreducible. Let \(\Delta (\omega )\) be the projective cover of \(L(\omega )\) in the category of \(S(\delta )\)-modules. We caution the reader that while \(L(\omega )\) will depend only on the chamber coweight \(\omega \) (as in [25]), the module \(\Delta (\omega )\) will depend not just on \(\omega \) but also on the positive system \(p(\Phi _{\succ \delta })\).
Theorem 17.1
Let \(\omega \) be a chamber coweight. There is a short exact sequence
Proof
As in the proof of Lemma 14.4, we may assume without loss of generality that \((\omega _+,\omega _-)\) is a minimal pair for \(\delta \).
By Lemma 16.1, there is a short exact sequence
for some projective \(S(\delta )\)-module C.
As C is cuspidal, \({\text {Res}}_{\omega _+\omega _-}C=0\). By adjunction, this implies that \({\text {Ext}}^i(q^2\Delta (\omega _+)\circ \Delta (\omega _-),C)=0\). From the long exact sequence obtained by applying \({\text {Hom}}(-,C)\) to (17.1), we therefore get an isomorphism
By Lemma 13.3 and adjunction,
which is zero since \(\Delta (\omega _+)\otimes \Delta (\omega _-)\) is a projective \(S(\omega _+)\otimes S(\omega _-)\)-module. From the long exact sequence obtained by applying \({\text {Hom}}(\Delta (\omega _-)\circ \Delta (\omega _+),-)\) to (17.1), we therefore have a surjection from \({\text {End}}(\Delta (\omega _-)\circ \Delta (\omega _+))\) onto \({\text {Hom}}(\Delta (\omega _-)\circ \Delta (\omega _+),C)\).
Again we apply Lemma 13.3 and adjunction to obtain
By Theorem 18.3 this is isomorphic to \({\mathbb {Q}}[x,y]\) with x and y in degree 2. Concentrating our attention to degree zero, we obtain \({\text {End}}(C)_0\cong {\mathbb {Q}}\). Therefore C is indecomposable.
The module \(L(\omega )\) is by construction a quotient of \(\Delta (\omega _-)\circ \Delta (\omega _+)\). Since it is cuspidal, the same argument that produced the isomorphism (17.2) yields an isomorphism
Therefore \(L(\omega )\) is a quotient of C. Since C is an indecomposable projective \(S(\delta )\)-module, it must be that C is the projective projective cover of \(L(\omega )\). \(\square \)
Corollary 17.2
Let \(\omega \) be a chamber coweight. Then \([\Delta (\omega )]\in {\mathbf {f}}\) and when specialised to \(q=1\) is equal to \(h_\omega \otimes t\).
Proof
This is immediate from Theorems 18.2 and 17.1. \(\square \)
As a consequence we also obtain the following theorem, which also appears in [25].
Theorem 17.3
The set of all modules \(L(\omega )\), as \(\omega \) runs over the chamber coweights adapted to the convex order \(\prec \), is a complete list of the cuspidal irreducible representations of \(R(\delta )\).
Proof
Corollary 17.2 shows that the modules \(\Delta (\omega )\) are a complete set of indecomposable projective modules for \(S(\delta )\). In the last paragraph of the proof of Theorem 17.1, we showed that the module \(L(\omega )\) is a quotient of \(\Delta (\omega )\). We also proved that \(L(\omega )\) is simple in Lemma 14.4. Therefore the set of such \(L(\omega )\) is a complete set of irreducible cuspidal representations of \(R(\delta )\). \(\square \)
Let \(\{n_\omega \}_{\omega \in \Omega }\) be a sequence of natural numbers. Lemma 14.2 shows that the induced product
is independent of the order of the factors.
Now we know the modules \(L(\omega )\) are pairwise nonisomorphic, we can use the same argument as in Theorem 14.3 to obtain a natural isomorphism
If \(\{m_\omega \}_{\omega \in \Omega }\) and \(\{n_\omega \}_{\omega \in \Omega }\) are two sequences of natural numbers then there is a natural inclusion
which, under the isomorphism (17.3) is the tensor product of the natural inclusions
If \(\omega \) is a chamber coweight and \(\lambda \) is a partition of n, we define
where \(S^\lambda \) is the Specht module for \(S_n\).
Let \(\underline{\lambda }=\{\lambda _\omega \}_{\omega \in \Omega }\) be a multipartition. Then we define
Define the multi-Littlewood-Richardson coefficients by
where \(c_{\lambda \mu }^\nu \) is the ordinary Littlewood-Richardson coefficient, which we take to be zero if \(|\nu |\ne |\lambda |+|\mu |\).
Theorem 17.4
The family of modules \(L({\underline{\lambda }})\) enjoy the following properties under induction and restriction:
Proof
This follows from the observation above that the inclusions (17.4) and (17.5) are equivalent under the isomorphism (17.3), together with the known formulae for the induction and restriction of Specht modules for the inclusions \(S_m\times S_n\rightarrow S_{m+n}\). \(\square \)
As a particular case of Theorem 17.4, we have
18 Homological modules
See the beginning of the previous section for a discussion of the inductive structure of the arguments in this section.
Theorem 18.1
Let \(\alpha \) be an indivisible positive root. Let \(\Delta \) and L be \(S(\alpha )\)-modules with \(\Delta \) projective. Then for all \(i>0\),
We remind readers that these Ext groups are taken in the category of \(R(\alpha )\)-modules which makes this result nontrivial.
Proof
Let \((\beta ,\gamma )\) be a minimal pair for \(\alpha \). If \(\alpha \) is a real root, then by the inductive hypothesis applied to Theorem 18.2 and Corollary 17.2, there exist projective \(S(\beta )\) and \(S(\gamma )\)-modules, \(\Delta _\beta \) and \(\Delta _\gamma \) such that \([\Delta _\beta ][\Delta _\gamma ]\ne q^{\beta \cdot \gamma }[\Delta _\gamma ][\Delta _\beta ]\). Therefore in the short exact sequence of Lemma 16.1, C is a nonzero direct sum of copies of \(\Delta (\alpha )\).
If \(\alpha \) is imaginary, then without loss of generality assume that \(\Delta \) is indecomposable projective, hence isomorphic to \(\Delta (\omega )\) for some \(\omega \). Then we use the short exact sequence of Theorem 17.1 and so in all cases we have a short exact sequence
and it suffices to prove that \({\text {Ext}}^i(C,L)=0\) for all cuspidal \(R(\alpha )\)-modules L.
By adjunction there is an isomorphism
Lemma 12.3 shows that \({\text {Res}}_{\gamma \beta }L\) is a \(S(\gamma )\otimes S(\beta )\)-module. Thus by inductive hypothesis we know that this Ext group is zero.
On the other hand, the group \({\text {Ext}}^{i-1}(q^{-\beta \cdot \gamma }\Delta _\beta \circ \Delta _\gamma ,L)\) is zero by adjunction and the cuspidality of L.
Now consider the short exact sequence (18.1) and apply \({\text {Hom}}(-,L)\) to get a long exact sequence of Ext groups. In the long exact sequence the group \({\text {Ext}}^i(C,L)\) is sandwiched between two groups which we have shown to be zero, hence must be zero itself. \(\square \)
Theorem 18.2
Let \(\alpha \) be a real root. Inside \({\mathbf {f}}^*_{{\mathbb {Z}}((q))}\) we have \([\Delta (\alpha )]=E_\alpha \).
Proof
By Theorem 18.1, \(\langle [\Delta (\alpha )],[L(\alpha )]\rangle =1\). We know that \(\Delta (\alpha )\) only has \(L(\alpha )\) appearing as a composition factor, and by Theorem 9.1, \([L(\alpha )]=E_\alpha ^*\). Therefore \(\Delta (\alpha )\) is a scalar multiple of \(E_\alpha \). By [19, Proposition 38.2.1], the automorphisms \(T_i\) preserve \((\cdot ,\cdot )\), hence \(\langle E_\alpha , E_\alpha ^* \rangle =1\) and the scalar is 1. \(\square \)
Theorem 18.3
Let \(\alpha \) be a real root. The endomorphism algebra of \(\Delta (\alpha )\) is isomorphic to \({\mathbb {Q}}[z]\), where z is in degree two.
Proof
As \(\Delta (\alpha )\) is the projective cover of \(L(\alpha )\) which is the unique simple \(S(\alpha )\)-module, the dimension of \({\text {End}}(\Delta (\alpha ))\) is equal to the multiplicity of \(L(\alpha )\) in \(\Delta (\alpha )\). Theorems 9.1 and 18.2 tell us that \([\Delta (\alpha )]=E_\alpha \) and \([L(\alpha )]=E_\alpha ^*\). Since \(E_\alpha ^*=(1-q^2)E_\alpha \), we have \(\dim {\text {End}}(\Delta (\alpha ))=(1-q^2)^{-1}\).
There is an injection from the centre of \(S(\alpha )\) into \({\text {End}}(\Delta (\alpha ))\). By Lemma 15.2, there is an injection from \({\mathbb {Q}}[z]\) into \({\text {End}}(\Delta (\alpha ))\). A dimension count shows that this injection must be a bijection, as required. \(\square \)
Corollary 18.4
Let \(\alpha \) be a positive real root. Then the algebras \(S(\alpha )\) and \({\mathbb {Q}}[z]\) are graded Morita equivalent.
Proof
The module \(\Delta (\alpha )\) is a projective generator for the category of \(S(\alpha )\)-modules and its endomorphism algebra is \({\mathbb {Q}}[z]\). \(\square \)
19 Standard imaginary modules
Lemma 19.1
Let \(d\le 0\) be an integer and let \(\omega \) and \(\omega '\) be two chamber coweights. Then
Proof
Since \(\Delta (\omega )\) is a projective \(S(\delta )\)-module, the dimension of \({\text {Hom}}(\Delta (\omega ),\Delta (\omega '))\) is equal to the multiplicity of \(L(\omega )\) in \(\Delta (\omega ')\).
We have
and by Lemma 18.1, the bases \(\{\Delta (\omega )\}\) and \(\{L(\omega )\}\) are dual bases for the subspace of \({\mathbf {f}}_\delta \) spanned by the cuspidal modules. Therefore
which shows the desired properties of the multiplicities. \(\square \)
Lemma 19.2
The module \({\underset{\omega \in \Omega }{\bigcirc }} \Delta (\omega )^{\circ n_\omega }\) is a projective object in the category of \(S(n\delta )\)-modules.
Remark 19.3
We choose an arbitrary ordering of the factors in \(\circ _{\omega \in \Omega }\Delta (\omega )^{\circ n_\omega }\). Lemma 19.4 below shows that this choice of ordering is immaterial.
Proof
Let L be a semicuspidal \(R(n\delta )\)-module. Therefore \({\text {Res}}_{\delta ,\ldots ,\delta }L\) is a \(S(\delta )\otimes \cdots \otimes S(\delta )\)-module. By adjunction
and since each \(\Delta (\omega )\) is a projective \(S(\delta )\)-module, this \({\text {Ext}}^1\) group is trivial, as required. \(\square \)
Lemma 19.4
Let \(\omega \) and \(\omega '\) be two chamber coweights. Then \(\Delta (\omega )\circ \Delta (\omega ')\cong \Delta (\omega ')\circ \Delta (\omega )\).
Proof
We assume that \(\omega \ne \omega '\) as otherwise the result is trivial. By Lemma 19.1 and a computation using adjunction and the Mackey filtration, we compute \({\text {End}}(\Delta (\omega )\circ \Delta (\omega '))_0\cong {\mathbb {Q}}\). Hence \(\Delta (\omega )\circ \Delta (\omega ')\) is indecomposable. By Lemma 19.2 the module \(\Delta (\omega )\circ \Delta (\omega ')\) is a projective \(S(2\delta )\)-module which surjects onto \(L(\omega )\circ L(\omega ')\), hence is the projective cover of \(L(\omega )\circ L(\omega ')\) in the category of \(S(2\delta )\)-modules. By Lemma 14.2, \(L(\omega )\circ L(\omega ')\cong L(\omega ')\circ L(\omega )\), hence their projective covers are isomorphic. \(\square \)
Theorem 19.5
Let \(\{m_\omega \}_{\omega \in \Omega }\) and \(\{n_\omega \}_{\omega \in \Omega }\) be two collections of natural numbers with \(\sum _\omega m_\omega =\sum _\omega n_\omega \) and let \(d\le 0\) be an integer. Then
Proof
The Mackey filtration for \({\text {Res}}_{\delta ,\ldots ,\delta }({\circ _{\omega \in \Omega }}\Delta (\omega )^{\circ n_\omega })\) has \((\sum _{\omega }n_\omega )!\) nonzero subquotients, each a tensor product of projective \(S(\delta )\)-modules where the factor \(\Delta (\omega )\) appears \(n_\omega \) times.
Therefore the filtration splits, and by Lemma 19.1 and adjunction, the Hom space under question is zero unless \(m_\omega =n_\omega \) for all \(\omega \) and \(d=0\). Furthermore in this case its dimension is \(\prod _w n_\omega !\).
Since \(\circ _{\omega \in \Omega }\Delta (\omega )^{\circ n_\omega }\) is a projective \(S(n\delta )\)-module and \(\circ _{\omega \in \Omega }L(\omega )^{\circ n_\omega }\) is a quotient of \(\circ _{\omega \in \Omega }\Delta (\omega )^{\circ n_\omega }\), every endomorphism of \(\circ _{\omega \in \Omega }L(\omega )^{\circ n_\omega }\) lifts to an endomorphism of \(\circ _{\omega \in \Omega }\Delta (\omega )^{\circ n_\omega }\). From the dimension counts in the previous paragraph and (17.3), this lift is unique in degree zero and hence we get an algebra isomorphism
So the result follows from (17.3). \(\square \)
For a multipartition \(\underline{\lambda }=\{\lambda _\omega \}_{\omega \in \Omega }\) where each \(\lambda _w\) is a partition of \(n_\omega \), we define
Theorem 19.6
The modules \(\Delta (\lambda )\) behave in the following way under induction and restriction.
Proof
The proof is the same as that of Theorem 17.4 \(\square \)
Let \(f_\lambda \) be the dimension of the Specht module \(S^\lambda \) and for a multipartition \({\underline{\lambda }}=\{\lambda _\omega \}_{\omega \in \Omega }\), let \(f_{\underline{\lambda }}=\prod _{\omega } f_{\lambda _w}\).
As a \({\mathbb {Q}}[S_n]\)-module, \({\mathbb {Q}}[S_n]\) decomposes as \({\mathbb {Q}}[S_n]=\oplus _{\lambda } (S^\lambda ) ^{\oplus f_\lambda }\). Therefore we obtain the decomposition
Lemma 19.7
Let \(\underline{\lambda }\) be a multipartition of n. The module \(\Delta (\underline{\lambda })\) is indecomposable.
Proof
From the decomposition (19.1) we obtain inclusions
Comparing dimensions shows that these inclusions are isomorphisms in degree zero. Therefore \({\text {End}}(\Delta (\underline{\lambda }))_0\) is isomorphic to \({\mathbb {Q}}\), hence \(\Delta ({\underline{\lambda }})\) is indecomposable. \(\square \)
Lemma 19.8
If \({\underline{\lambda }}\ne \underline{\mu }\), then \(\Delta ({\underline{\lambda }})\) is not isomorphic to any grading shift of \(\Delta (\underline{\mu })\).
Proof
Let \(i\le 0\) be an integer. The inclusions in (19.2) are all isomorphisms in degrees less than or equal to zero. Therefore \({\text {Hom}}(\Delta (\lambda ),\Delta (\mu ))_i=0\) and thus \(\Delta (\lambda )\) is not isomorphic to \(q^i \Delta (\mu )\). Similarly \(\Delta (\mu )\) is not isomorphic to \(q^i \Delta (\lambda )\). \(\square \)
Theorem 19.9
The set \(\Delta (\underline{\lambda })\) is a complete set of indecomposable projective \(S(n\delta )\)-modules.
Proof
The module \(\Delta (\underline{\lambda })\) is a direct summand of \(\circ _{\omega \in \Omega } \Delta (\omega )^{\circ n_\omega }\) which is projective by Lemma 19.2, hence \(\Delta ({\underline{\lambda }})\) is projective. Lemmas 19.7 and 19.8 ensure that the set \(\{\Delta ({\underline{\lambda }})\}\) is an irredundant set of indecomposable projective \(S(n\delta )\)-modules, up to a grading shift. The number of indecomposable projective \(S(n\delta )\)-modules is equal to the number of irreducible semicuspidal \(R(n\delta )\)-modules. This number is known by Theorem 8.9, hence we have found all of the indecomposable projectives. \(\square \)
Theorem 19.10
The set \(\{L(\underline{\lambda })\}_{{\underline{\lambda }}\vdash n}\) is a complete set of self-dual irreducible \(S(n\delta )\)-modules.
Proof
The set \(\Delta (\underline{\lambda })\) is a complete set of indecomposable projectives, so the set \({\text {hd}}\Delta (\underline{\lambda })\) is a complete set of irreducible \(S(n\delta )\)-modules. Since \(\Delta (\underline{\lambda })\) surjects onto \(L(\underline{\lambda })\), the set \({\text {hd}}(L(\underline{\lambda }))\) is a complete set of irreducible \(S(n\delta )\)-modules. So it suffices to prove that \(L(\underline{\lambda })\) is irreducible.
Let X be a simple submodule of \(L(\underline{\lambda })\). Then X is semicuspidal so is of the form \({\text {hd}}(L(\underline{\mu }))\) for some multipartition \(\underline{\mu }\). Therefore we get a nonzero morphism from \(L(\underline{\mu })\) to \(L(\underline{\lambda })\). From the decomposition \({\underset{\omega \in \Omega }{\bigcirc }} L(\omega )^{\circ n_\omega }\cong \oplus _{\underline{\lambda }}\Delta ({\underline{\lambda }})^{\oplus f_{{\underline{\lambda }}}}\) we obtain inclusions
Comparing dimensions shows that these inclusions are equalities and hence all morphisms from \(L(\underline{\mu })\) to \(L(\underline{\lambda })\) are either zero or isomorphisms. Hence \(L(\underline{\lambda })\) must be irreducible, as required. The self-duality of \(L({\underline{\lambda }})\) is immediate from the self-duality of \(L(\omega )\) and (5.2). \(\square \)
Theorem 19.11
Let \({\underline{\lambda }}\) and \({\underline{\mu }}\) be two multipartitions. Then
Proof
In the course of proving Theorem 19.10, the module \(\Delta (\underline{\lambda })\) was shown to be the projective cover of the irreducible module \(L(\underline{\lambda })\) in the category of \(S(n\delta )\)-modules. This takes care of the \(i=0\) case.
Now suppose that \(i>0\). Since \(\Delta (\underline{\lambda })\) is a direct summand of \(\circ _{\omega \in \Omega }\Delta (\omega )^{\circ n_\omega }\), it suffices to show that
The module \({\text {Res}}_{\delta ,\ldots ,\delta }L(\underline{\mu })\) has all composition factors a tensor product of cuspidal \(R(\delta )\)-modules. The result now follows from adjunction and Theorem 18.1. \(\square \)
Corollary 19.12
Let \({\underline{\lambda }}\) and \(\underline{\mu }\) be two multipartitions. Then \(\langle [\Delta ({\underline{\lambda }})],[L({\underline{\mu }})]\rangle =\delta _{{\underline{\lambda }},{\underline{\mu }}}\).
20 The imaginary part of the PBW basis
We now follow [2] and define the imaginary root vectors. For comparison with their paper, we note that our q is their \(q^{-1}\). We will not be able to cite results from [2] since they only work with convex orders of a particular type. The aim of this section is to describe a purely algebraic construction of the PBW basis. We will prove that this algebraic construction agrees with the one coming from KLR algebras in Theorem 24.4.
Let \(\omega \) be a chamber coweight adapted to \(\prec \). We first define elements \(\psi _n^\omega \) by
Before we continue, we show that the \(\psi _n^\omega \) lie in a commutative subalgebra of \({\mathbf {f}}\).
Theorem 20.1
If \(L_1\) and \(L_2\) are irreducible semicuspidal representations of \(R(n_1\delta )\) and \(R(n_2\delta )\) respectively, then \(L_1\circ L_2\cong L_2\circ L_1\).
Proof
The modules \(L_1\) and \(L_2\) are both direct summands of modules of the form \(\circ _{\omega } L(\omega )^{\circ n_\omega }\). The space of homomorphisms between two modules of this form has already been computed to be concentrated in degree zero. Therefore \({\text {Hom}}(L_1,L_2)\) is concentrated in degree zero. By the same argument as in the proof of Lemma 14.2, the R-matrices \(r_{L_1,L_2}\) and \(r_{L_2,L_1}\) are inverse isomorphisms. \(\square \)
Corollary 20.2
The subalgebra of \({\mathbf {f}}\) spanned by all semicuspidal representations of \(R(n\delta )\) is commutative.
Lemma 20.3
Let \(\omega \in \Omega \) and \(n\in {\mathbb {N}}\). There exist semicuspidal representations X and Y of \(R(n\delta )\) with \([X]-[Y]=\psi _n^\omega \).
Proof
The same argument as in the proof of Lemma 16.1 shows that we can take X and Y to be the cokernel and kernel of a map from \(q^2\Delta (\omega _+)\circ \Delta (n\delta -\omega _+)\) to \(\Delta (n\delta -\omega _+)\circ \Delta (\omega _+)\).
\(\square \)
Corollary 20.4
The elements \(\psi ^\omega _n\) commute with each other.
Now we return to defining the imaginary part of the PBW basis and recursively define elements \(P_n^\omega \) by \(P_0^\omega =1\) and
Let \(\lambda =(\lambda _1\ge \lambda _2\ge \cdots )\) be a partition and let \(t\ge \ell (\lambda )\) be an integer. We define
By Corollary 20.4 the entries in this matrix all commute with each other so there is no ambiguity in the definition of the determinant.
The elements \(P^\omega _n\) here should be thought of as playing the role of the complete symmetric functions in the ring of all symmetric functions. This determinental definition shows that the elements \(S^\omega _\lambda \) are playing the role of the Schur functions. This point of view makes it clear that the definition of \(S^\omega _\lambda \) does not depend on t.
Let \(\pi =(\beta _1^{m_1},\ldots ,\underline{\lambda },\ldots ,\gamma _1^{n_1})\) be a root partition. The PBW basis element \(E_\pi \) is defined to be
This agrees with the definition in [5] for the special convex orders which they use.
21 MV polytopes
Definition 21.1
Let M be an \(R(\nu )\)-module. The MV polytope of M, denoted P(M), is the convex hull of the set
Let \(\omega \) be a chamber coweight. The \(\omega \)-face of a polytope P is defined to be the intersection of P with the plane spanned by \(\omega _+\) and \(\omega _-\). For a general polytope, this construction is a cross-section. We choose to call it a face because of the following result.
Proposition 21.2
Suppose that \(\omega \) is adapted to the convex order \(\prec \). If \(L(\pi )\) is a simple module for some root partition \(\pi \) such that the support of \(\pi \) is contained in the span of \(\omega _-\) and \(\omega _+\), then the \(\omega \) face of \(P(L(\pi ))\) is a (possibly degenerate) 2-face of \(P(L(\pi ))\).
Definition 21.3
Let \(\lambda \) be the functional on the span of \(\omega _-\) and \(\omega _+\) such that \(\lambda (\omega _+)=1\) and \(\lambda (\omega _-)=-1\). The width of the \(\omega \)-face of a polytope P is equal to the maximum value of \(\lambda (p)-\lambda (q)\) where p and q are two points in the \(\omega \)-face of P.
Example 21.4
The width of the \(\omega \)-face of \(P(L(n\delta -\omega _+))\) is n.
We know this because a MV polytope is completely determined by its 2-faces, which are MV polytopes for rank two root systems.
In the rest of this section, we fix a choice of chamber coweight \(\omega \) adapted to \(\prec \). Without loss of generality, we may assume that our convex order is of the form of Example 3.6.
Note that in our labelling of the irreducible semicuspidal modules for \(R(\delta )\) by multipartitions, there are choices involved. Namely replacing \(r_{L(\omega ),L(\omega )}\) by its negative results in replacing the partition \(\lambda _\omega \) by its transpose. We make a choice of sign in \(r_{L(\omega ),L(\omega )}\) such that \(L_\omega ((2))\) has \(\omega \)-width 2.
The reason that such a choice is always possible is that the module \(L(\omega ')\circ L(\omega '')\) will only have \(\omega \)-width at least two if \(\omega =\omega '=\omega ''\) and by the Tingley-Webster classification, there exists a unique MV polytope for \(2\delta \) of \(\omega \)-width 2. It must thus come from one of the summands of \(L(\omega )\circ L(\omega )\) and we may replace our R-matrix with its negative if necessary to ensure that this summand is the one indexed by the partition (2).
This means that the \(\omega \)-face of the MV polytope for \(L(\omega ,(1^2))\) is
Proposition 21.5
Let \(\lambda \) be a partition and \(\omega \) be a chamber coweight. The module \(L_\omega (\lambda )\) has \(\omega \)-width 1 if and only if \(\lambda =(1^n)\).
Proof
We prove this proposition by an induction on n. The case \(n=1\) is trivial and the case \(n=2\) is true by the choice of normalisation of the R-matrix.
Note that for \(\omega '\ne \omega \), the module \(L(\omega ')\) has \(\omega \)-width zero. Therefore the \(\omega \)-width of \({\underset{x\in \Omega }{\bigcirc }} L(\lambda _x)\) is equal to the \(\omega \)-width of \(L(\lambda _w)\).
Therefore by induction we know exactly how many \(\omega \)-faces of modules of the form \({\underset{x\in \Omega }{\bigcirc }} L(\lambda _x)\) with \(|\lambda _w|<n\) have width less than or equal to one. By [25] this comprises all MV polytopes of \(\omega \)-width less than or equal to one except for one polytope of \(\omega \)-width one. Therefore there exists some partition \(\mu \vdash n\) for which \(L_\omega (\mu )\) has \(\omega \)-width one.
The restriction \({\text {Res}}_{k\delta ,(n-k)\delta }L(\mu )\) can only have composition factors \(L_1\otimes L_2\) where \(L_1\) and \(L_2\) have \(\omega \)-width at most one. These restrictions are given by the Littlewood-Richardson rule (17.4). So by induction the only option is \({\text {Res}}_{k\delta ,(n-k)\delta }L(\mu )\cong L_\omega (1^k)\otimes L_\omega (1^{n-k})\) which for \(n>2\) forces \(\mu =(1^n)\), completing the proof. \(\square \)
Theorem 21.6
Proof
We perform an expansion in the dual PBW basis
Then
We consider the algorithm of Sect. 11 which teaches us how to write the product \(E_\sigma E_\pi \) in terms of the PBW basis.
Let \(\pi _k\) be the smallest root appearing in \(\pi \). If \(\pi \ne \omega _+\) then as \(\omega _+-\pi _k\in {\mathbb {N}}I\), it must be that \(\pi _k \preceq \omega _-\). Therefore at all stages in applying the algorithm for writing \(E_\sigma E_\pi \) in terms of the PBW basis, any term \(E_{\gamma _1}\cdots E_{\gamma _l}\) which appears has \(\gamma _l\preceq \omega _- \prec \delta \). Therefore no purely imaginary terms in the PBW basis can appear, and as \([L_\omega (1^n)]\) is orthogonal to all PBW elements which are not purely imaginary, \(c_{\sigma ,\pi }=0\) for such \(\pi \).
So we may assume \(\pi =\omega _+\).
Let \(\sigma _k\) be the smallest root appearing in \(\sigma \). Suppose that \(\sigma _k\) is not of the form \(m\delta +\omega _-\). Then \(\sigma _k\prec \omega _-\). At the first stage of applying our algorithm, up to two terms \(E_{\gamma _1}\cdots E_{\gamma _l}\) appear. One term has \(\gamma _l=\sigma _k\prec w_-\) while the other term, if it exists, has \(\gamma _l=\sigma _l+\omega _+\) which is also less than \(\omega _-\), since by convexity it is less than \(\omega _+\) and we know all roots between \(\omega _+\) and \(\omega _-\). By the same argument as in the previous paragraph, \(c_{\sigma ,\pi }=0\) in this case too.
Therefore, when \(c_{\sigma ,\pi }\ne 0\), all roots that appear in \(\sigma \) are all in the span of \(\omega _-\) and \(\omega _+\). This implies that every irreducible subquotient of \({\text {Res}}_{n\delta -\omega _+,\omega _+} L_\omega (1^n)\) is of the form \(L(\sigma )\otimes L(\omega _+)\) for some such root partition \(\sigma \).
The largest root appearing in \(\sigma \) is at most \(\delta \) as \(L_\omega (1^n)\) is cuspidal. Therefore \(\sigma =(\underline{\lambda },m\delta -\omega _+)\) for some multipartition \(\underline{\lambda }\) and positive integer m.
The \(\omega \)-face of \(P(L(m\delta -\omega _+))\) has width m. Therefore the \(\omega \)-face of \(P(L(\sigma ))\) has width at least m. As the \(\omega \)-face of \(P(L(\sigma ))\) is a subset of the \(\omega \)-face of \(P(L_\omega (1^n))\) which as width one, \(m=1\).
Now by Theorem 17.4,
Therefore the only option for \(\underline{\lambda }\) is \(1^n\) at \(\omega \) and zero elsewhere, and furthermore \(L(1^n_\omega ,\omega _-)\otimes L(\omega _+)\) must appear with multiplicity one, completing the proof. \(\square \)
Lemma 21.7
Let \(\omega \) be a chamber coweight and \(\alpha =\omega _-\). There is a short exact sequence
Proof
Theorem 10.1 tells us that \(L(\omega ,\alpha )\) is the head of the module \(L(\omega )\circ L(\alpha )\) and that every other subquotient of \(L(\omega )\circ L(\alpha )\) is cuspidal. Therefore there is a short exact sequence
for some cuspidal \(R(\alpha +\delta )\)-module X. Since the head of \(L(\omega )\circ L(\alpha )\) is known, Lemma 7.5 implies that \([X]\in q{\mathbb {N}}[q]E_{\alpha +\delta }^*.\) Taking duals there is a short exact sequence
We now consider
The restriction has two nonzero pieces in its Mackey filtration. The module \(L(\alpha )\otimes L(\omega )\) appears as a quotient and we use Lemma 14.4 to identify the submodule as \(L(\alpha )\otimes (L(\delta -\alpha )\circ L(\alpha ))\).
Now we consider
where we have used the adjunction (6.2) and Lemma 14.4 to reach this isomorphism. Therefore there is a unique (up to scalar) morphism from \(L(\alpha )\circ L(\omega )\) to \(L(\omega )\circ L(\alpha )\) in degree 2, and the only other possible morphisms are in degree zero from the other term in the Mackey filtration. When comparing this with \([X]\in q{\mathbb {N}}[q]E_{\alpha +\delta }^*\), the only option is that \(X\cong q L(\alpha +\delta )\), as required. \(\square \)
Lemma 21.8
Let \(\omega \) be a chamber coweight and \(\alpha =\omega _-+n\delta \) for some natural number n. Then there are short exact sequences
Proof
We prove the existence of these short exact sequences by an induction on n. The case \(n=0\) for the first sequence is Lemma 21.7. First we prove the existence of the first sequence for some \(n>0\), assuming that both sequences are known for lesser values of n.
As in the proof of Lemma 21.7, we have a short exact sequence
where \([X]\in q{\mathbb {N}}[q]E_{\alpha +\delta }^*\), and we wish to study
The Mackey filtration of \({\text {Res}}_{\alpha ,\delta }(L(\omega )\circ L(\alpha ))\) has two nonzero pieces. The module \(L(\alpha )\otimes L(\omega )\) appears as a quotient, and to understand the submodule, we need to first understand \({\text {Res}}_{\alpha -\delta ,\delta }L(\alpha )\).
By Lemma 12.3, we can write
for some polynomials \(g_x(q)\in {\mathbb {N}}[q,q^{-1}]\) which satisfy \(g_x(q)=g_x(q^{-1})\) since restriction commutes with duality.
For \(x\in \Omega \), let \(C_x\) be the projective \(S(\alpha -\delta )\)-module which appears in the short exact sequence of Lemma 16.1:
We compute
Therefore \(C_x\cong g_x(q)\Delta (\alpha )\).
If \(x\ne \omega \) then we can compute the value of \([C_x]\) after specialising \(q=1\) in \({\mathbf {f}}\) to obtain \(g_x(1)\) is 0 or 1, which forces \(g_x(q)\) to be 0 or 1.
For \(x=\omega \), we use the inductive hypothesis applied to the second short exact sequence to conclude that \(g_\omega (q)=q+q^{-1}\). Therefore \({\text {Res}}_{\alpha ,\delta } L(\omega )\circ L(\alpha -\delta )\) has a submodule isomorphic to \(q(L(\omega )\circ L(\alpha -\delta ) )\otimes L(\omega )\).
By the inductive hypothesis this module receives a map from \(q^2 L(\alpha )\otimes L(\omega )\) and hence there exists a morphism from \(L(\alpha )\circ L(\omega )\) to \(L(\omega )\circ L(\alpha )\) of degree two. In fact this argument shows us we know even more, namely that all other morphisms between these modules are of degree zero. So the same argument as in Lemma 21.7 allows us to conclude \(X\cong qL(\alpha +\delta )\), as required.
Now we deduce the second short exact sequence from the first. By Lemma 16.1, there exists a short exact sequence
where C is a projective \(S(\alpha +\delta )\)-module, hence isomorphic to f(q) copies of \(\Delta (\alpha +\delta )\) for some \(f(q)\in {\mathbb {N}}[q,q^{-1}]\). The same argument computing pairings as above shows that f(q) is equal to the multiplicity of \(L(\alpha )\otimes L(\omega )\) in \({\text {Res}}_{\alpha ,\delta }L(\alpha +\delta )\). The computation in \({\mathbf {f}}\) specialised at \(q=1\) shows \(f(1)=2\), and since \(f(q)=f(q^{-1})\), we have \(f(q)=q^i+q^{-i}\) for some \(i\in {\mathbb {Z}}\).
The first exact sequence gives us a morphism from \(qL(\alpha +\delta )\) to \(L(\omega )\circ L(\alpha )\) which by adjunction induces a nonzero morphism \({\text {Res}}_{\alpha ,\delta }L(\alpha +\delta )\rightarrow L(\alpha )\otimes L(\omega )\). Therefore \(i=1\), as required. \(\square \)
Proposition 21.9
Let k and l be positive integers. There is a short exact sequence
Proof
This proof proceeds by an induction. By Theorem 10.1, the module \(L_\omega (1^{k+1})\circ L(l\delta -\omega _+)\) surjects onto \(A( L_\omega (1^{k+1}),L(l\delta -\omega _+))\) and all other subquotients are of the form \(X_{\underline{\lambda },m}^i=q^iA(L(\underline{\lambda }),L((l+m)\delta -\alpha ))\) for some \(m>0\) and \(\underline{\lambda }\) a multipartition of \(k+1-m\).
Setting \(n=k+1-m\), the following computation is straightforward as there is only one nonzero piece in the Mackey filtration.
Note that if \(X_{\underline{\lambda },m}^i\) is a subquotient of \(L_\omega (1^{k+1})\circ L(l\delta -\omega _+)\) then \(L(\underline{\lambda })\otimes L((l+m)\delta -\omega _+)\) must appear as a subquotient of this restriction. Immediately we see that \(\lambda _\omega =(1^n)\) and \(\lambda _x=0\) for all other chamber coweights x.
Consider a subquotient of the form \(X_{\underline{\lambda },m}^i\) with \(\underline{\lambda }\ne 0\). Then by inductive hypothesis we know all that there is only a cuspidal subquotient of \( L_\omega (1^m)\circ L(l\delta -\omega _+)\) when \(m=1\). Furthermore this cuspidal subquotient appears with multiplicity q, which completes the proof in this case.
So now turn our attention to the remaining case when \(n=0\). The module \(L_\omega (1^{k+1})\circ L(l\delta -\omega _+)\) has \(\omega \)-width \(l+1\) and the module \(L((l+m)\delta -\omega _+)\) has \(\omega \)-width \(l+m\). Therefore \(m=1\). The result now follows from Lemma 21.8. \(\square \)
22 Inner product computations
For any natural number n and chamber coweight \(\omega \), define \(e_n^\omega =[L_\omega (1^n)]\).
Lemma 22.1
Let \(\omega \) be a chamber coweight and \(\{n_x\}_{x\in \Omega }\) a collection of natural numbers with sum n. Then
Proof
By Theorem 13.1, we assume without loss of generality that our convex order is as in Example 3.6.
By definition \(\psi _n^\omega =E_{n\delta -\omega _+} E_{\omega _+}-q^2 E_{\omega _+} E_{n\delta -\omega _+}\). Since \({\text {Res}}_{\omega _+,n\delta -\omega _+}L=0\) for any semicuspidal representation L, we have
The terms in the product all commute so without loss of generality we may assume that the \(r(e_{n_\omega }^\omega )\) term is last.
Each term appearing in the product of the \(r(e_{n_x}^x)\)’s is a product of terms \(y\otimes z\) with y of degree at most \(\delta \) and z of degree at least \(\delta \). Since we need a term of degree \((n\delta -\omega _+,\omega _+)\), the only option is that exactly one of the terms does not have degree \((n_x\delta ,0)\).
That particular term will have degree \((n_x-\omega _+,\omega _+)\). Now for \(r_{n_x-\omega _+,\omega _+}(e_{n_x}^x)\) to not be zero, it must be that \({\text {Res}}_{n_x\delta -\omega _+,\omega _+}L_x(1^{n_x})\ne 0\) and hence the restriction \({\text {Res}}_{n_x\delta -\omega _+,\omega _+}L(x)^{\circ n_x}\) is also not zero. By a Mackey argument this implies that \({\text {Res}}_{\omega _-,\omega _+}L(x)\ne 0\). By Lemma 12.3 there is an injection \(L(\omega _-)\otimes L(\omega _+)\rightarrow {\text {Res}}_{\omega _-,\omega _+}L(x)\) and so by adjunction there is a nonzero map from \(L(\omega _-)\circ L(\omega _+)\) to L(x). By Theorem 17.3 \(x=\omega \).
Now Theorem 21.6 and Proposition 21.9 tell us that
Therefore
Since \({\text {Res}}_{\delta ,(n-1)\delta -\omega _+}\Delta (n\delta -\omega _+)=0\), there is only one possible term which can be nonzero, it only occurs when \(n_x=0\) for all \(x\ne \omega \) and \(j=n_\omega \). The resulting inner product is easily evaluated to \((-q)^{n-1}\). \(\square \)
Lemma 22.2
For \(n\ge 0\), we have
Proof
From the definition of \(P_n^\omega \),
Since each \(L_x(1^{n_x})\) is semicuspidal, the only relevant terms in \(r(e_{n_x}^x)\) are of bidegree \((k\delta ,l\delta )\) for some \(k,l\in {\mathbb {N}}\), and all these terms are known by (17.6). Therefore
This can easily be computed by an induction on n together with Lemma 22.1. \(\square \)
23 Symmetric functions
Let \(\Lambda \) be the Hopf algebra of symmetric functions. We consider it over the ground ring \({\mathbb {Z}}[q,q^{-1}]\). It is isomorphic to \({\mathbb {Z}}[q,q^{-1}][h_1,h_2,\ldots ]\) where \(h_n\) is the complete symmetric function. Let \(s_\lambda \) be the Schur function indexed by the partition \(\lambda \). Let \((\cdot ,\cdot )\) denote the usual inner product on \(\Lambda \) for which the Schur functions form an orthonormal basis. We denote the coproduct on \(\Lambda \) by \(\Delta \).
Let B be the subalgebra of \({\mathbf {f}}^*\) generated by the elements \(e_{n}^\omega \). For \(x\in B\) we define \(r_\delta (x)\in B\otimes B\) to be the sum of all terms in r(x) of bidegree \((a\delta ,b\delta )\).
Lemma 23.1
There is an isomorphism of Hopf algebras \(\psi \,{:}\,\Lambda ^{\otimes \Omega }\!\rightarrow \!B\) with
where the coproduct on B is \(r_\delta \).
Proof
This is immediate from Theorem 17.4. \(\square \)
Define an algebra homomorphism \(\varphi \,{:}\,\Lambda ^{\otimes \Omega }\!\rightarrow \!{\mathbf {f}}\) by
That such a homomorphism exists is because the \(h_{n_\omega }\) freely generate \(\Lambda ^{\otimes \Omega }\) as a commutative algebra and Corollary 20.4 which implies that the \(P_{n_\omega }^\omega \) lie in a commutative subalgebra of \({\mathbf {f}}\).
Lemma 23.2
For all \(x,y\in \Lambda ^{\otimes \Omega }\) we have
Proof
Lemma 22.2 establishes this formula in the special case when \(x=P_n^\omega \). To deduce the general case from this particular case, we use \((xy,z)=(x\otimes y,z)\) and
where in the last step we used Lemma 23.1. \(\square \)
Corollary 23.3
Let \(\omega \) and \(\omega '\) be two chamber coweights and let \(\lambda \) and \(\mu \) be partitions. Then \(\langle S_\lambda ^\omega ,[L_{\omega '}(\mu )] \rangle = \delta _{\omega \omega '}\delta _{\lambda \mu }\).
Proof
The Schur functions are orthonormal. \(\square \)
Theorem 23.4
Let \({\underline{\lambda }}=\{\lambda _\omega \}_{\omega \in \Omega }\) be a purely imaginary root partition. Then
Proof
The nondegeneracy of \((\cdot ,\cdot )\) together with Lemma 23.2 implies that \(\varphi \) is injective. By Lemmas 20.3 and 8.2, the image of \(\varphi \) lies in the subspace of \({\mathbf {f}}_{{\mathbb {Z}}((q))}^*\) spanned by the semicuspidal modules. A dimension count shows that the image is precisely the span of the semicuspidal modules. Therefore \(\Delta _\omega ({\underline{\lambda }})\) is a linear combination of the elements \(S_{\underline{\mu }}\).
The pairings in Corollary 19.12 and 23.3 force \(\Delta (\underline{\lambda })=S_{\underline{\lambda }}\). \(\square \)
24 Standard modules
The nil Hecke algebra \(NH_n\) is the algebra R(ni) for any \(i\in I\). It is well known that the nil Hecke algebra is a matrix algebra over its centre, see for example [24, Proposition 2.21]. In particular, there is an isomorphism
where each \(x_i\) is in degree two.
Let \(e_n\) be a primitive idempotent in \(NH_n\).
Theorem 24.1
Let \(\alpha \) be a real root. There is an isomorphism \({\text {End}}(\Delta (\alpha )^{\circ n})\cong NH_n\).
Proof
The proof of [4, §3] works in this generality without any change. \(\square \)
For any positive real root \(\alpha \) and any positive integer n, we define the divided power standard module \(\Delta (\alpha )^{(n)}\) to be
Lemma 24.2
Let \(\alpha \) be a real root and n a positive integer. Then
Proof
We compute by adjunction
The module \({\text {Res}}_{\delta ,\ldots ,\delta }L(\alpha )^{\circ n}\) has a composition series with n! subquotients, each isomorphic to some \(q^jL(\alpha )^{\otimes n}\) and \([{\text {Res}}_{\delta ,\ldots ,\delta }L(\alpha )^{\circ n}]=[n]!E_\alpha ^*\otimes \cdots \otimes E_\alpha ^*\). So by Theorem 18.1, for \(i>0\) we have
while for \(i=0\) we also use the fact that \(\Delta (\alpha )\) is the projective cover of \(L(\alpha )\) in the category of \(S(\alpha )\)-modules to obtain
Since \(\Delta (\alpha )^{\circ n}\cong q^{{n \atopwithdelims ()2}} [n]!\Delta (\alpha )^{(n)}\), we obtain the desired result. \(\square \)
Let \(\pi =(\beta _1^{m_1},\ldots ,\beta _k^{m_k},\underline{\lambda },\gamma _l^{n_l},\ldots ,\gamma _1^{n_1})\) be a root partition. We define the corresponding standard module to be
Also define
Proposition 24.3
Let \(\pi \) and \(\sigma \) be two root partitions. Then
Proof
Let \(\pi =(\beta _1^{m_1},\cdots ,\gamma _1^{n_1})\). Then by adjunction,
and by Theorem 10.1, \({\text {Res}}_\pi \nabla (\sigma )=0\) unless \(\pi \le \sigma \).
On the other hand, by the adjunction (6.2),
and so again using Theorem 10.1, \({\text {Res}}_\sigma \Delta (\pi )=0\) unless \(\sigma \le \pi \).
Thus the only case to consider is when \(\sigma \sim \pi \). Remember that this means that \(\sigma \) and \(\pi \) agree except for the multipartition they contain. Let \({\underline{\lambda }}\) be the multipartition in \(\pi \) and \({\underline{\mu }}\) be the multipartition in \(\sigma \).
By Theorem 10.1,
Therefore
where the tensor product is over all real roots \(\alpha \).
The result now follows from Lemma 24.2 and Theorem 19.11. \(\square \)
Theorem 24.4
Let \(\pi \) be a root partition. The class of the standard module \(\Delta (\pi )\) is the PBW monomial \(E_\pi \), defined algebraically in (20.1).
Proof
This follows from Theorems 18.2 and 23.4. \(\square \)
Proposition 24.3 proves that the classes of the standard modules \(\Delta (\pi )\) and the proper standard modules \(\overline{\Delta }(\sigma )\) are orthogonal under \((\cdot ,\cdot )\). Therefore the class of each proper standard module is an element of the dual PBW basis. So we have categorified both the PBW and dual PBW basis.
A module M is said to have a \(\Delta \)-flag if it has a sequence of submodules \(0=M_0\subseteq M_1\subseteq \cdots \subseteq M_{n-1}\subseteq M_n=M\) such that each subquotient \(M_{i+1}/M_i\) is isomorphic to \(q^m\Delta (\pi )\) for some integer m and some root partition \(\pi \).
Theorem 24.5
Let M be a finitely generated \(R(\nu )\)-module such that \({\text {Ext}}^1(M,\nabla (\pi ))=0\) for all root partitions \(\pi \). Then M has a \(\Delta \)-flag. Furthermore \([M:\Delta (\pi )]=\dim {\text {Hom}}(M,\nabla (\pi ))\).
Proof
This is a standard argument, for example see [4, Theorem 3.13]. \(\square \)
As a consequence we obtain the following BGG reciprocity for KLR algebras.
Theorem 24.6
Let \(\pi \) be a root partition and let \({\mathcal {P}}(\pi )\) be the projective cover of \(L(\pi )\). Then \({\mathcal {P}}(\pi )\) has a \(\Delta \)-flag. For any root partition \(\sigma \) the multiplicity \([{\mathcal {P}}(\pi ):\Delta (\sigma )]\) is equal to the multiplicity \([\overline{\Delta }(\sigma ):L(\pi )]\).
Proof
Since \({\mathcal {P}}(\pi )\) is finitely generated and projective it satisfies the hypotheses of Theorem 24.5 and hence has a \(\Delta \)-flag. Furthermore the multiplicity of the module \(\Delta (\sigma )\) in the flag is
As \({\mathcal {P}}(\pi )\) is the projective cover of \(L(\pi )\), the dimension of this homomorphism space is equal to the multiplicity \([\nabla (\sigma )^\circledast :L(\pi )]\). By duality \([\nabla (\sigma ):L(\pi )]=[\overline{\Delta }(\sigma ):L(\pi )^\circledast ]\) and since \(L(\pi )\cong L(\pi )^\circledast \) we are done. \(\square \)
Theorem 24.7
The PBW basis (20.1) is a basis of \({\mathbf {f}}\) as a \({\mathbb {Z}}[q,q^{-1}]\)-module.
Proof
By Theorem 24.6 and Theorem 10.1(3), the matrix expressing the set \(\{[\Delta (\pi )]\}\) in terms of the basis \(\{[{\mathcal {P}}(\pi )]\}\) is upper-triangular, with ones along the diagonal. Therefore the set \(\{[\Delta (\pi )]\}\) is a basis of \({\mathbf {f}}\) as a \({\mathbb {Z}}[q,q^{-1}]\)-module. \(\square \)
Remark 24.8
This is a generalisation, with a different proof, of a result of [5].
Proposition 24.9
With respect to the PBW basis, the bar involution is unitriangular.
Proof
By Proposition 24.3, the PBW basis is dual to the basis \([\nabla (\pi )]\) under the pairing \(\langle \cdot ,\cdot \rangle \). It suffices to prove that the bar involution on \({\mathbf {f}}^*\) is unitriangular with respect to this basis. Since each \(\nabla (\pi )\) is an induction product of self-dual simples up to an overall grading shift, it is easy to see that the bar involution is unitriangular by Theorem 10.1(3). \(\square \)
Once we have that the bar-involution is unitriangular, it is straightforward to show that there exists a unique basis \(b_\pi \) of \({\mathbf {f}}\) which is bar-invariant and for which
where \(c_{\pi \sigma }\in q{\mathbb {Z}}[q]\). Theorem 24.10 below shows that the basis \(\{b_\pi \}\) is the canonical basis, providing an algebraic characterisation of the canonical basis.
Thus from Theorem 24.6 and the fact that the indecomposable projective modules categorify the canonical basis, we obtain the following positivity result.
Theorem 24.10
The change of basis matrix from the canonical basis to a PBW basis is unitriangular with off diagonal entries lying in \(q{\mathbb {N}}[q]\).
Proof
The fact that the coefficients are all nonnegative is from Theorem 24.6 and the fact that the indecomposable projective modules categorify the canonical basis. That the coefficients lie in \(q{\mathbb {Z}}[q]\) follows from Lemma 7.5. \(\square \)
This positivity result is new in affine type. In finite type this result is [18, Corollary 10.7] for particular convex orders and for all convex orders is due to Kato and the author independently in [8, 22].
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McNamara, P.J. Representations of Khovanov–Lauda–Rouquier algebras III: symmetric affine type. Math. Z. 287, 243–286 (2017). https://doi.org/10.1007/s00209-016-1825-4
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DOI: https://doi.org/10.1007/s00209-016-1825-4