Generalized Involution Models of Projective Reflection Groups

Abstract

The main motivation of this work was to investigate the generalized involution models of the projective reflection groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we have been led to consider and solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p′, q′, n) are isomorphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if $G(r,p,1,n)\mathop{\cong}G(r,1,p,n)$. We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd.

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The details for this work are provided in our paper (F. Caselli and E. Marberg, Isomorphisms, automorphisms, and generalized involution models of projective reflection groups, Israel J. Math., 50 pp., in press).

A model for a finite group G is a set $\{\lambda _{i}: H_{i} \rightarrow \mathbb{C}\}$ of linear characters of subgroups of G, such that the sum of induced characters $\sum _{i}\mathrm{Ind}_{H_{i}}^{G}(\lambda _{i})$ is equal to the multiplicity-free sum of all irreducible characters ∑ _{ψ ∈ Irr(G)} ψ. Models are interesting because they lead to interesting representations in which the irreducible representations of G live. This is especially the case when the subgroups H _i are taken to be the stabilizers of the orbits of some natural G-action.

FormalPara Example 1

Let G = G(r, n) be the group of complex n × n matrices with exactly one nonzero entry, given by an rth root of unity, in each row and column. Assume r is odd. Then G acts on its symmetric elements by $g: X\mapsto gXg^{T}$, and the distinct orbits of this action are represented by the block diagonal matrices of the form

$$\displaystyle{X_{i}\stackrel{\mathrm{def}}{=}\left (\begin{array}{ll} J_{2i}&0 \\ 0 &I_{n-2i} \end{array} \right ),}$$

where J _n denotes the n × n matrix with ones on the anti-diagonal and zeros elsewhere. Write H _i for the stabilizer of X _i in G. The elements of H _i preserve the standard copy of $\mathbb{C}^{2i}$ in $\mathbb{C}^{n}$, inducing a map $\pi _{i}: H_{i} \rightarrow \mathrm{ GL}_{2i}(\mathbb{C})$. If $\lambda _{i}\stackrel{\mathrm{def}}{=}\mathrm{det} \circ \pi _{i}$ then $\{\lambda _{i}: H_{i} \rightarrow \mathbb{C}\}$ is a model for G(r, n) [2, Theorem 1.2].

The following definition of Bump and Ginzburg [5] captures the salient features of this example. Let ν be an automorphism of G with ν ² = 1. Then G acts on the set of generalized involutions

$$\displaystyle{\mathcal{I}_{G,\nu }\stackrel{\mathrm{def}}{=}\{\omega \in G:\omega ^{-1} =\nu (\omega )\}}$$

by the twisted conjugation $g:\omega \mapsto g \cdot \omega \cdot \nu (g)^{-1}$. We write

$$\displaystyle{C_{G,\nu }(\omega )\stackrel{\mathrm{def}}{=}\{g \in G: g \cdot \omega \cdot \nu (g)^{-1} =\omega \}}$$

to denote the stabilizer of $\omega \in \mathcal{I}_{G,\nu }$ under this action, and say that a model $\{\lambda _{i}: H_{i} \rightarrow \mathbb{C}\}$ is a generalized involution model (or GIM for short) with respect to ν if each H _i is the stabilizer C _G, ν (ω) of a generalized involution $\omega \in \mathcal{I}_{G,\nu }$, with each twisted conjugacy class in $\mathcal{I}_{G,\nu }$ contributing exactly one subgroup. The model in Example 1 is a GIM with respect to the inverse transpose automorphism of G(r, n).

In [13, 14], the second author classified which finite complex reflection groups have GIMs. Subsequently, the first author discovered an interesting reformulation of this classification, which suggests that these results are most naturally interpreted in the broader context of projective reflection groups. These groups were introduced in [7] and further studied, for example, in [4, 6, 8]. They include as an important special case an infinite series of groups G(r, p, q, n) defined as follows.

For positive integers r, p, n with p dividing r, let G(r, p, n) denote the subgroup of G(r, n) consisting of the matrices whose nonzero entries, multiplied together, form an (r∕p)th root of unity. Apart from 34 exceptions, the irreducible finite complex reflection groups are all groups G(r, p, n) of this kind. The projective reflection group G(r, p, q, n) is defined as the quotient

$$\displaystyle{G(r,p,q,n)\stackrel{\mathrm{def}}{=}G(r,p,n)/C_{q}}$$

where C _q is the cyclic subgroup of scalar n × n matrices of order q. Note that for this quotient to be well-defined we must have $C_{q} \subset G(r,p,n)$, which occurs precisely when q divides r and pq divides rn. Observe also that G(r, n) = G(r, 1, n) and G(r, p, n) = G(r, p, 1, n).

There is an interesting notion of duality for projective reflection groups; by definition, the projective reflection group dual to G = G(r, p, q, n) is $G^{{\ast}}\stackrel{\mathrm{def}}{=}G(r,q,p,n)$. This notion of duality has been crucial in the study of some aspects of the invariant theory of these groups in [7] and in the construction of other type of models in [6, 8]. The starting point of the present collaboration is now the following theorem which reformulates the main result of [13].

FormalPara Theorem 1

The complex reflection group G = G(r,p,1,n) has a GIM if and only if $G\mathop{\cong}G^{{\ast}}$ ; i.e., if and only if $G(r,p,1,n)\mathop{\cong}G(r,1,p,n)$ .

FormalPara Remark 1

Explicitly, G has a GIM if and only if (i) n ≠ 2 and GCD(p, n) = 1 or (ii) n = 2 and either p or r∕p is odd; this is the statement of [13, Theorem 5.2].

Deducing this theorem from [13, Theorem 5.2] is straightforward, given our next main result. Let r, n be positive integers and let p, p′, q, q′ be positive divisors of r such that pq = p′q′ divides rn. The following result simplifies and extends [7, Theorem 4.4].

FormalPara Theorem 2

The projective reflection groups G(r,p,q,n) and G(r,p′,q′,n) are isomorphic if and only if either (i) GCD (p,n) = GCD (p′,n) and GCD (q,n) = GCD (q′,n) or (ii) n = 2 and the numbers p + p′ and q + q′ and $\frac{r} {pq}$ are all odd integers.

As a corollary, we can say precisely when the group G(r, p, q, n) is “self-dual” as in Theorem 1.

FormalPara Corollary 1

The projective reflection group G = G(r,p,q,n) is isomorphic to its dual G ^∗ = G(r,q,p,n) if and only if either (i) GCD (p,n) = GCD (q,n) or (ii) n = 2 and $\frac{r} {pq}$ is an odd integer.

On seeing Theorem 1 one naturally asks whether for arbitrary projective reflection groups the property of having a GIM is equivalent to self-duality. Theorem 2 allows us to attack this question directly; its answer turns out to be false, and the rest of our results are devoted to clarifying which groups G(r, p, q, n) have GIMs. The following theorem completely solves this problem in the often pathological case n = 2.

FormalPara Theorem 3

The projective reflection group G(r,p,q,2) has a GIM if and only if (r,p,q) = (4,1,2) or $G(r,p,q,2)\mathop{\cong}G(r,q,p,2)$ .

FormalPara Remark 2

By Theorem 2, the condition $G(r,p,q,2)\mathop{\cong}G(r,q,p,2)$ holds if and only if (i) p and q have the same parity or (ii) $\frac{r} {pq}$ is an odd integer.

A few notable differences between complex reflection groups and projective reflection groups complicates the task of determining the existence of GIMs, and in the case n ≠ 2 our classification is incomplete. For example, the groups G(r, p, q, n) occasionally can have conjugacy class-preserving outer automorphisms. The fact that the groups G(r, p, n) never have such automorphisms [15, Proposition 3.1] was the source of a significant reduction in the proof of [13, Theorem 5.1] which is no longer available in many cases of interest. Nevertheless, by carrying out a detailed analysis of the conjugacy classes and automorphisms of G(r, p, q, n), we are able to prove the following theorem.

FormalPara Theorem 4

Let G = G(r,p,q,n) and assume n ≠ 2.

• (1) If GCD (p,n) = 1 then G has a GIM if q or n is odd.

• (2) If GCD (p,n) = 2 then G has a GIM only if q is even.

• (3) If GCD (p,n) = 3 then G has a GIM if and only if (r,p,q,n) is

  $$\displaystyle{(3,3,3,3)\text{ or }(6,3,3,3)\text{ or }(6,6,3,3)\text{ or }(6,3,6,3).}$$

• (4) If GCD (p,n) = 4 then G has a GIM only if r ≡ p ≡ q ≡ n ≡ 4 (mod  8).

• (5) If GCD (p,n) ≥ 5 then G does not have a GIM.

In arriving at this result, we prove a useful criterion for determining conjugacy in G(r, p, n) and give an explicit description of the automorphism group of G(r, p, q, n). We note as a corollary that the theorem provides a complete classification when q or n is odd. This shows that projective reflection groups which are not self-dual may still possess GIMs.

FormalPara Corollary 2

Let G = G(r,p,q,n) and assume n ≠ 2 and (r,p,q,n) is not one of the four exceptions (3,3,3,3) or (6,3,3,3) or (6,6,3,3) or (6,3,6,3). If q or n is odd, then G has a GIM if and only if GCD (p,n) = 1.

Combining Theorems 2 and 4 shows that to completely determine which projective reflection groups G(r, p, q, n) have GIMs, it remains only to consider groups of the form

$$\displaystyle{G(2r,1,2q,2n)\quad \text{or}\quad G(2r,2,2q,2n)\quad \text{or}\quad G(8r + 4,4,8q + 4,8n + 4).}$$

(Of course we only need to consider the first two types when 2n > 2.) We also have some conjectures concerning which of these groups should have GIMs.

This research continues a line of inquiry taken up by a number of people in the past few decades. Researchers originally considered involution models, which are simply GIMs defined with respect to the identity automorphism. Inglis et al. described an elegant involution model for the symmetric group in [9] (which is precisely the model in Example 1 when r = 1). In his doctoral thesis, Baddeley [3] classified which finite Weyl groups have involution models. Vinroot [16] extended this classification to show that the finite Coxeter groups with involution models are precisely those of type A _n , BC _n , D _2n+1, F ₄, H ₃, and I ₂(m). In extending this classification to reflection groups, it is natural to consider generalized involution models, since only groups whose representations are all realizable over the real numbers can possess involution models. Adin et al. [2] constructed a GIM for G(r, n) extending Inglis, Richardson, and Saxl’s original model for S _n , which provides the starting point of [13, 14].

As mentioned at the outset, these sorts of classifications are interesting because they lead to interesting representations. We close our contribution with some recent evidence of this phenomenon. The model in Example 1 with r = 1 gives rise via induction to a representation of S _n on the vector space spanned by its involutions. This representation turns out to have a simple combinatorial definition [1, Sect. 1.1], which surprisingly makes sense mutatis mutandis for any Coxeter group. The generic Coxeter group representation we get in this way corresponds to an involution model (in the finite cases) in precisely types A _n , H ₃, and I ₂(2m + 1). What’s more, recent work of Lusztig and Vogan [11, 12] and Vogan [10] indicates that this representation is the specialization of a Hecke algebra representation which for Weyl groups is expected to have deep connections to the unitary representations of real reductive groups.