Abstract
D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han.
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1 Introduction
Since the times of Euler, Gauss, and Jacobi properties and formulas of the coefficients \(a_{n} \left( r\right) \) of integral powers of Euler products, now known as powers of the Dedekind eta function \(\eta ^{r} \), have been studied [1, 2, 18, 23]. These involve pentagonal numbers, partition numbers, and the Ramanujan tau-function [19] as the most prominent examples. Non-vanishing properties are of particular interest, e. g. the Lehmer conjecture [17] addressing \(r = 24\). A still outstanding result was given by Serre [23] in 1985 for r positive and even. The sequence \(a_{n} \left( r\right) \) is lacunary if and only if \(r \in \left\{ 2, 4, 6, 8, 10, 14, 26\right\} \).
In this paper, we significantly improve results of Kostant [16] and Han [9] on the non-vanishing of the coefficients. Kostant proved that \(a_{n} \left( m^{2} - 1\right) \ne 0\) for \(m \ge \max \left\{ 4, n\right\} \) by using Macdonald’s fundamental theory on affine root systems [18] and the identification of \(\left| a_{n} \left( m^{2} - 1\right) \right| \) with the dimension of some special Lie algebras. In 2010, Han extended Kostant’s result to \(r \in {\mathbb {R}}\) applying the Nekrasov–Okounkov [20] hook length formula.
The starting point is the Dedekind eta function. Its powers \(\eta ^{r}\) (\(r \in {\mathbb {Z}}\)) are one of the most well-known and most studied functions in mathematics [2, 3, 6, 14, 18, 22].
where \(q := e^{2\pi i\tau }\), \(\mathop {\mathrm{Im}}\left( \tau \right) >0\). The coefficients are special values of the D’Arcais polynomials \(P_n(x)\) [4, 5, 13, 21, 23, 24]. Let \(z \in {\mathbb {C}}\). Let
These polynomials are special cases of recursively defined polynomials \(P_n^g(x)\) associated with a normalized, i. e. \(g\left( 1\right) =1\), arithmetic function \(g: {\mathbb {N}} \longrightarrow {\mathbb {N}}\). In this paper, we are mainly interested in the cases \(g(n)= \sigma (n)\) and \(g(n) = n^l\), where \(l\in {\mathbb {N}}_0\). Let \(P_0^g(x):=1\) and
for \(n\ge 1\). Let \(g(n)= \sigma (n):=\sum _{d \mid n} d\), then \(P_n(x) = P_n^{\sigma }(x)\) and \(a_n(r) = P_n^{\sigma }(-r)\). Let \(g(n)=n\), then it can be shown that \(P_n^g(x)\) is proportional to an associated Laguerre polynomial, see [10]. As a special case of Theorem 6 and Corollary 3, we obtain our main results with respect to the Dedekind eta function. We prove in this paper, see Sect. 4:
Theorem
Let \(\kappa =15\). For all \(z \in {\mathbb {C}}\) and \(n \in {\mathbb {N}}\) with \( \vert z \vert > \kappa (n-1)\), we obtain the growth condition
Corollary
Let \(\vert z \vert > \kappa (n-1)\), then \(P_n(z) \ne 0\).
Let for example \(z = -(10^6-1)\). Then, the result of Kostant [16, Theorem 4.28] implies that \(P_n(z) \ne 0\) for all \(n \le 10^3\). Our Theorem implies that this is already true for \(n\le 6 \cdot 10^4\).
The work of Kostant and Nekrasov–Okounkov led to new non-vanishing results towards coefficients of the powers of the Dedekind eta function. In this work, we utilize properties of the D’Arcais polynomials to obtain new results beyond their results. Actually, we even get an unexpected new type of result in the context of Kostant’s study of complex simple Lie algebras \({\mathfrak {g}}\). We denote by \(\lfloor \,\,\, \rfloor \) the Gauss bracket.
Application
Let \( {\mathfrak {g}}\) be a complex simple Lie algebra (to simplify we exclude the types \(A_1,A_2,G_2\)). Let \(h^{\vee }\) be the dual Coxeter number and
Let \(C_n \subset \wedge ^n {\mathfrak {g}}\) be the span of all 1-dimensional subspaces of the form \(\wedge ^n {\mathfrak {a}}\), where \({\mathfrak {a}} \subset {\mathfrak {g}}\) is any n-dimensional abelian subalgebra of \({\mathfrak {g}}\) (see also Sect. 2). Then, \(\dim \, C_n \ne 0\) if and only if \( 1 \le n \le h^{\vee }\). Further \(\dim C_n = a_n (\dim \, {\mathfrak {g}})\) (result of Kostant [16]). Our Theorem implies:
2 Kostant’s formula
We recall a result of Kostant [16] involving alternating sums of \(\dim \,V_{\lambda }\), where \(V_{\lambda }\) is the irreducible module with the highest weight \(\lambda \) of a complex simple Lie algebra. The highest weight uniquely determines the representation \(\pi _{\lambda }\) up to equivalence and also the Casimir operator. We obtain growth results and significantly improve the non-vanishing results obtained by Kostant.
Let \({\mathfrak {g}}\) be a complex simple Lie algebra. We choose a simply connected compact group K, such that \({\mathfrak {k}} = \mathop {\mathrm{Lie}}\, K\) is a compact form of \({\mathfrak {g}}\). Let \(T \subset K\) be a maximal torus and \({\mathfrak {h}}:= i \, \mathop {\mathrm{Lie}}\, T\). We identify \({\mathfrak {h}}\) with its dual with respect to the Killing form such that \(\Delta \subset {\mathfrak {h}}\) for the set of roots for the pair \(({\mathfrak {h}}_{{\mathbb {C}}},{\mathfrak {g}})\). Here, \(\Delta ^{+}\) denotes a set of positive roots and \({\mathfrak {h}}^{+}\) the corresponding Weyl chamber.
Let \(D \subset {\mathfrak {h}}^{+}\) be the set of dominant integral forms of \({\mathfrak {h}}\). Then, every \(\lambda \in D\) corresponds to an irreducible representation
with the highest weight \(\lambda \). For the following (including notation), we refer to Theorem 3.1 in [15] and Theorem 0.1 in [16]. Let \(\rho \) be the Weyl element and \(a_P:= \exp (2 \pi \, i \, 2 \, \rho )\).
Theorem 1
(Kostant [15]) For any \(\lambda \in D\), the value of the character \(\chi _{\lambda }\) of \(\pi _{\lambda }\) evaluated at \(a_P\) is an element of \(\{-1,0,1\}\). Let \(\mathop {\mathrm{Cas}}(\lambda )\) be the scalar value taken by the Casimir element of \(V_{\lambda }\). Then,
We are interested in the vanishing properties of the coefficients \(a_n=a_n(\dim \, K)\) defined by
Let \(W_f\) denote the affine Weyl group acting on \({\mathfrak {h}}\). Let \(\psi \in \Delta ^{+}\) be the highest root. Then,
is a fundamental domain. Let \(\sigma \in W_f\), then \(A_{\sigma }:= \sigma (A_1)\) is called an alcove. An alcove is dominant if \(A_{\sigma } \subset {\mathfrak {h}}^{+}\). We put
Let \(\rho \) be the Weyl element and \(\sigma \in W_f^{+}\) then
Theorem 2
(Kostant [16]) Let \(\lambda \in D\). Then, \(\chi _{\lambda }(a_P) \in \{-1,1\}\) if and only if
In this case, \( \chi _{\lambda }(a_P) = (-1)^{l(\sigma )}, \) where \(l(\sigma )\) is the length of \(\sigma \). The coefficients \(a_n= a_n(\dim K)\) are given by
As Kostant already indicated [16, Sect. 4.6]: one major difficulty in using formula (11) to determine \(a_n\) is the cancellation in the sums due to the alternation of signs. He discovered that if \(n \le h^{\vee }\) (dual Coxeter number), then this alternation does not occur, and \((-1)^n a_n\) can be identified with the dimension of certain algebras, see Kostant [16], Theorem 4.23. Here, we recall one of these identifications and an application towards non-vanishing of the coefficients \(a_n\).
Let \(n \in {\mathbb {N}}_0\). Then, we denote by \(C_n \subset \wedge ^n {\mathfrak {g}}\) the span of all 1-dimensional subspaces of the form \(\wedge ^n {\mathfrak {a}}\), where \({\mathfrak {a}} \subset {\mathfrak {g}}\) is any n-dimensional abelian subalgebra of \({\mathfrak {g}}\). Then, \(C_n \ne 0 \Leftrightarrow n \le M\), where M is the maximal dimension of a commutative subalgebra of \( {\mathfrak {g}}\). Malcev computed M for each simple Lie algebra \({\mathfrak {g}}\). We give a complete list. The cases \(A_{m-1}\) and \(G_2\) had been worked out in [16]. Note that \(M < h^{\vee } \Leftrightarrow {\mathfrak {g}}\) is of type \(A_1, A_2\) and \(G_2\) and \(M = h^{\vee }\) otherwise.
Theorem 3
(Kostant [16]) Let \({\mathfrak {g}}\) be a complex simple Lie algebra. Let \(n \le h^{\vee }\). Then,
These coefficients are zero if and only if \(M \le n \le h^{\vee }\). Hence, \(a_n = 0\) if and only if the \({\mathfrak {g}}\) type is \(A_1\) and \(n=2\), or \(A_2\) and \(n=3\), or \(G_2\) and \(n=4\).
The direct application of our theorem given in the introduction (see also Theorem 6) leads to new insights and improvement of the results of Kostant (we also refer to [16] Theorem 4.28 and [9] Theorem 1.6).
Theorem 4
Let \({\mathfrak {g}}\) be a complex simple Lie algebra. Let \(\lambda ^{\sigma }= \frac{\sigma (2 \rho )}{2} - \rho \), where \(\rho \) is the Weyl element and \(\sigma \in W_f^{+}\). Let \( \pi _{\lambda ^{\sigma }}: K \longrightarrow \mathop {\mathrm{Aut}}(V_{\lambda ^{\sigma }})\) be the corresponding irreducible representation. We denote by \(l(\sigma )\) the length of the Weyl group element. Let \(\dim \, K > \kappa \, (n-1)\), where \(\kappa =15\). Then,
Corollary 1
Let \(\dim \, K > \kappa \, (n-1)\). Then,
Example
Let \({\mathfrak {g}}\) be of type \(A_{m-1}\). Let \(m=10^3\) then \(\dim \, K = 10^6-1\). Then, Kostant’s result implies that (14) is true for \(n \le 10^3\). Theorem 4 implies that (14) is already true for
3 The Nekrasov–Okounkov hook length formula
Almost at the same time as Kostant published his paper, Nekrasov and Okounkov [9, 20, 24] discovered a new type of hook length formula.
We follow the introduction given in [12]. Random partitions and the Seiberg–Witten theory lead to an identity between a sum over products of partition hook lengths [7, 8] and the coefficients of complex powers of Euler products [11, 21, 23], which is essentially a power of the Dedekind eta function.
Let \(\lambda \) be a partition of n denoted by \(\lambda \vdash n\) with weight \(|\lambda |=n\). We denote by \({\mathcal {H}}(\lambda )\) the multiset of hook lengths associated with \(\lambda \) and by \({\mathcal {P}}\) the set of all partitions. The Nekrasov–Okounkov hook length formula [9, Theorem 1.2] is given by
The identity (15) is valid for all \(z \in {\mathbb {C}}\). Our result in this context is the following.
Theorem 5
Let \(n \in {\mathbb {N}}\) and \(\kappa =15\). Let \(z \in {\mathbb {C}}\) and \( \vert z \vert > \kappa \, (n-1)\). Then,
This is a new type of growth condition related to the Nekrasov–Okounkov hook length formula.
Remark
a) Let z be a positive real number. Then, (15) implies:
b) Let z be a negative real number. Han observed [9, Theorem 1.6]: let \(z < 1-n^2\), then
If \(-z\ge 4\), then (18) is already true for \(z \le 1-n^2\).
Theorem 5 implies the following non-vanishing result.
Corollary 2
Let \(n \in {\mathbb {N}}\) and \(\kappa =15\). Let \(z \in {\mathbb {C}}\) and \( \vert z \vert > \kappa \, (n-1)\). Then,
Let z be a negative real number. Then, \(z < \kappa \, (1-n)\) implies
This is true, since the left hand side of (19) is a polynomial in z of degree n, which is non-vanishing for real z smaller than \(\kappa \, (1-n)\) and thus behaves like \(z^{n}\).
4 Growth conditions on D’Arcais-type polynomials \(P_n^g(x)\)
Recall the setting from the introduction. Let \(g:{\mathbb {N}}\rightarrow {\mathbb {N}}\) with \(g\left( 1\right) =1\) be a normalized arithmetic function. We further associate with g(n) a family of polynomials \(P_n^g(x)\) and a (shifted) generating function G(T) with positive radius R of convergence. We put \(P_0^g(x)=1\) and
for \(n\ge 1\). Let further
Theorem 6
Let a normalized arithmetic function g be given. Let \(P_n^g(x)\) be the associated polynomials and let G(T) be the generating function with positive radius of convergence. Then, for any constant \( \kappa >0\) with \(G(\frac{2}{\kappa }) \le \frac{1}{2}\) we have the following estimation.
if \(\left| x \right| > {\kappa } \left( n-1 \right) \), for all \(n \in {\mathbb {N}}\).
Proof
The proof will be given by induction on n. We start with \(n=1\). Let \(\left| x\right| >0\). Then,
Let us assume the theorem is true for \( 1 \le j \le n-1\). The induction step is based on the following inverse triangle inequality, employing (21)
We are allowed to assume for \(1\le j\le n-1\):
Iterating this inequality leads to
for all \(k=2,\ldots ,n\). Using this, we can now estimate the sum in (24), which is involved in the lower bound of \(\vert P_n^g(x)\vert \):
This leads to the crucial inequality
Estimating the sum and using the assumption from the theorem, we obtain
Note that \(\frac{2n-2}{\left| x\right| }<\frac{2}{\kappa }\). Since G is increasing on \(\left[ 0,R\right) \) as \(g\left( k\right) >0\) for all \(k\in {\mathbb {N}}\), the theorem is proven. \(\square \)
In particular, for the sum of divisors function we obtain:
Corollary 3
Let \(g=\sigma \) and \(\left| x \right| > 15 \left( n-1 \right) \) for \(n \ge 1\). Then,
Proof
We have to find an upper bound on \(G\left( q\right) =\sum _{k=1}^{\infty }\sigma \left( k+1\right) q^{k}\) . Let \(h\left( k\right) =\sigma \left( k\right) \) for \(1\le k\le 4\) and \(h\left( k\right) =\left( k+1\right) k\) for \(k\ge 4\) . Then, obviously \(\sigma \left( k\right) \le h\left( k\right) \) for all \(k\in {\mathbb {N}}\). This implies \(G\left( q\right) \le \sum _{k=1}^{\infty }h\left( k+1\right) q^{n}=F\left( q\right) \) for \(0\le q\le 1\le R\). The series F is now almost (except for the first 4 terms) the second derivative of the geometric series of q:
For \(q=\frac{2}{15}\), we obtain
The claim now follows from the previous theorem. \(\square \)
Taking more values of h equal to \(\sigma \) does not seem to yield a significant improvement any more. For example, taking \(h\left( k\right) =\sigma \left( k\right) \) for \(k\le 9\) would only allow us to take \(\kappa =14.76\).
The previous estimate on the growth of the polynomials \(P_{n}\left( x\right) \) has important implications.
Corollary 4
\(\displaystyle P_{n}\left( x\right) \ne 0\) for \(\left| x \right| > 15\left( n-1 \right) \), \(n\ge 1\).
Similarly, the theorem can be exploited to find uniform constants \(\kappa =\kappa _{m}\) only depending on a function \(h:{\mathbb {N}}\rightarrow {\mathbb {N}}\) for all functions \(g:{\mathbb {N}}\rightarrow {\mathbb {N}}\) that satisfy \(g\left( k\right) \le h\left( k\right) \) . One example is the following:
Corollary 5
Let \(m\in {\mathbb {N}}\cup \left\{ 0\right\} \) be fixed. Suppose \(g:{\mathbb {N}}\rightarrow {\mathbb {N}}\) satisfies
for all \(n\in {\mathbb {N}}\) . Then, for all such g the constant c in Theorem 6 can be chosen as
Proof
By our assumption, the power series \(G\left( q\right) =\sum _{k=1}^{\infty }g\left( k\right) q^{k}\) satisfies for \(0\le q< 1\le R\):
since the series is essentially the mth derivative of the geometric series in q. For \(\kappa _{m}=\frac{2}{1-\root m+1 \of {2/3}}\), we obtain \( \frac{1}{2}\) for \(q= \frac{2}{\kappa _{m}} \) in the series. \(\square \)
In the following, we list the values of \(\kappa _{m}\) for \(m=0,1,2\) and integer bounds on them. They are related to the interesting cases considered in [13]. Here, \(m=0\) leads to polynomials which have Stirling numbers of the first kind as their coefficients. The case \(m=1\) leads to associated Laguerre polynomials. And \(m=2\) leads to polynomials which can be considered as an upper bound of the D’Arcais polynomials \(P_n^{\sigma }(x)\).
Corollary 6
For \(m=0,1,2\), we obtain \(\kappa _{0}=\frac{2}{1-\frac{2}{3}}=6\), \(\kappa _{1}=\frac{2}{1-\sqrt{\frac{2}{3}}}<11\), and \(\kappa _{2}=\frac{2}{1-\root 3 \of {\frac{2}{3}}}<16\).
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Heim, B., Neuhauser, M. The Dedekind eta function and D’Arcais-type polynomials. Res Math Sci 7, 3 (2020). https://doi.org/10.1007/s40687-019-0201-5
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DOI: https://doi.org/10.1007/s40687-019-0201-5