Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups

For a positive integer $N$, $\mathfrak{S}_{N}$ is the symmetric group of degree $N$ and acts on $\mathbb{Z}^{N}$ by

$\displaystyle\begin{array}[]{cccccc}\mathfrak{S}_{N}&\curvearrowright&\mathbb{Z}^{N}&\xrightarrow{\simeq}&\mathbb{Z}^{N}&\\ \rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}&\\ \sigma&\curvearrowright&\lambda:=(\lambda_{1},\lambda_{2},\ldots,\lambda_{N})&\mapsto&\sigma.\lambda:=(\lambda_{\sigma^{-1}(1)},\lambda_{\sigma^{-1}(2)},\ldots,\lambda_{\sigma^{-1}(N)}).&\end{array}$

(3)

Let $\mathbb{C}[x_{1},x_{2},\ldots,x_{N}]$ be the ring of polynomials in $N$ independent variables $x_{1},x_{2},\ldots,x_{N}$ with complex numbers coefficients. The group $\mathfrak{S}_{N}$ also acts on the ring $\mathbb{C}[x_{1},x_{2},\ldots,x_{N}]$ by permutation of the variables $x_{i}\mapsto\sigma(x_{i}):=x_{\sigma(i)}$. We consider a subring

$$S_{N}:=\mathbb{C}[x_{1},x_{2},\ldots,x_{N}]^{\mathfrak{S}_{N}}:=\{f\in\mathbb{C}[x_{1},x_{2},\ldots,x_{N}]\mid\sigma(f)=f\>\>\text{for any}\>\>\sigma\in\mathfrak{S}_{N}\},$$

and we call elements of $S_{N}$ symmetric polynomials. For any positive integer $k$, the set

$$S_{N}^{k}:=\{f\in S_{N}\mid\mathrm{deg}(f)=k\}$$

is a finite-dimensional vector space over $\mathbb{C}$, and the subring $S_{N}$ is graded and has the following decomposition:

$$S_{N}=\bigoplus_{k\geq 0}S_{N}^{k}.$$

We denote the set of partitions of length $\leq N$ by

$$\mathcal{P}_{N}:=\left\{(\lambda_{1},\lambda_{2},\ldots,\lambda_{N})\in\mathbb{Z}^{N}\mid 0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{N}\right\}$$

and define the Monomial Symmetric Polynomial (MSP) by

$\displaystyle m_{\lambda}(x):=\sum_{\mu\in\mathfrak{S}_{N}.\lambda}x^{\mu},\quad\lambda\in\mathcal{P}_{N},$

where $\mathfrak{S}_{N}.\lambda:=\left\{\sigma.\lambda\mid\sigma\in\mathfrak{S}_{N}\right\}$ and $x^{\mu}:=x_{1}^{\mu_{1}}x_{2}^{\mu_{2}}\cdots x_{N}^{\mu_{N}}$. For any $\lambda\in\mathbb{N}^{N}$, the stabilizer subgroup of $\mathfrak{S}_{N}$ with respect to $\lambda$ is defined by $\mathfrak{S}_{N}^{\lambda}:=\left\{\sigma\in\mathfrak{S}_{N}\mid\sigma.\lambda=\lambda\right\}$. By the definition of MSP and

$$\left|\mathfrak{S}_{N}^{\lambda}\right|=\prod_{i\in\mathbb{N}}(\lambda[i]!),\quad\lambda[i]:=|\left\{j\mid\lambda_{j}=i\right\}|,$$

we obtain a well-known expression of the MSP:

$\displaystyle m_{\lambda}(x)=\prod_{i\in\mathbb{N}}\frac{1}{\lambda[i]!}\sum_{\sigma\in\mathfrak{S}_{N}}x^{\sigma.\lambda}.$

(4)

If $\lambda\in\mathbb{Z}^{N}$, then $m_{\lambda}(x)$ defines a $\mathfrak{S}_{N}$-invariant Laurent polynomial. For any partitions satisfying the condition $|\lambda|:=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{N}=k$, the set $\{m_{\lambda}(x)\}_{\lambda}$ forms a standard basis of the vector space $S_{N}^{k}$. If we consider some special partitions $\lambda=(k):=(0,0,\ldots,0,k)$ or $\lambda=(1^{k}):=(0,0,\ldots,0\overbrace{\rule{0.0pt}{9.0pt}1,1,\ldots,1}^{k})$, MSP $m_{\lambda}(x)$ become the $k$th power sum

$$m_{(k)}(x)=p_{k}(x):=\sum_{i=1}^{N}x_{i}^{k}$$

and the $k$th elementary symmetric polynomial

$$m_{(1^{k})}(x)=e_{k}(x):=\sum_{1\leq i_{1}<\cdots<i_{k}\leq N}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}},$$

respectively.

These symmetric polynomials and their variations are very fundamental and important in various fields such as multivariate special functions [10], [13], combinatorics [1], representation theory, harmonic analysis [3], [6], and even outside mathematics in mathematical physics and statistics [4], [12]. Not only the symmetric polynomials themselves, but also their special values, are equally fundamental and important. In fact, many classical special sequences like binomials coefficients, Stirling numbers, Fibonacci and Lucas numbers are expressed as special values of these symmetric polynomials. One of the most important and standard specialization of symmetric polynomials is the principal specializations

$$x=(1,q,q^{2},\ldots,q^{N-1}),\quad q\in\mathbb{C},$$

which are related to dimension formulas of irreducible representations for some groups or algebras and some enumeration formulas of various partitions.

In our paper, for $N=kn$ and a primitive complex $n$th root of unity $\zeta_{n}:=e^{\frac{2\pi\sqrt{-1}}{n}}$, we consider a specialization

$$\zeta_{(n,k)}:=(1,\zeta_{n},\zeta_{n}^{2},\ldots,\zeta_{n}^{kn-1})\in\mathbb{C}^{kn}$$

and study special values $m_{\lambda}(\zeta_{(n,k)})$ which appear naturally zonal spherical functions of Gelfand pairs for the complex reflection groups or arithmetic exponential sums. From a generating function for $m_{\lambda}(\zeta_{(n,k)})$ (see Proposition 2.1), these special values $m_{\lambda}(\zeta_{(n,k)})$ appear the generalized Waring’s formula [9] that is a formula to expand $e_{l}(x_{1}^{n},x_{2}^{n},\ldots,x_{n}^{n})$ by $\{e_{l}(x)\}_{l}$.

The MSP $m_{\lambda}(x)$ is a special case of the Macdonald polynomial $P_{\lambda}(x;q,t)$ (see [10, Chapter VI]) whose the principal specialization evaluate explicitly. However, since $m_{\lambda}(x)=P_{\lambda}(x;0,1)$, it is very hard to obtain explicit or simple expression of $m_{\lambda}(\zeta_{(n,k)})$ in general. Our main results give some explicit formulas and vanishing (or non-vanishing) properties of these special values for partitions

$$\Lambda_{n}^{k}:=\left\{(\lambda_{1},\lambda_{2},\ldots,\lambda_{kn})\in\mathbb{Z}^{kn}\mid 1\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{kn}\leq n\right\}$$

satisfying some conditions. First, we have the following results from some properties of MSP and Lemma 2.2.

On the other hand, from Dedekind’s theorem on group determinants for finite abelian groups, special values $m_{\lambda}(\zeta_{(n,k)})$ appear as coefficients of $k$th power of the group determinant a cyclic group $\mathbb{Z}/n\mathbb{Z}$. Also, in order to determine the multiplication table of the group from the group determinant, the coefficients of the terms of several types of the group determinant were obtained by Mansfield [11]. Hence, by applications of some results for group determinants, we have the second main results.

From the point of view of the group determinant, these results derive some non-vanishing properties for the terms of the group determinant.

The content of this paper is as follows. In Section 2, we mention a generating function of $m_{\lambda}(\zeta_{(n,k)})$ which is a specialization of the dual Cauchy kernel. From this generating function, we have integer and vanishing properties of $m_{\lambda}(\zeta_{(n,k)})$ (Theorem 1.2 (5) and (6)). We also prove a reduction formula of the sum over the symmetric group $\mathfrak{S}_{n}$ for a periodic function, which is a key step in the proof of Theorem 1.1 (1). Next, we introduce some fundamental results for the group determinant of finite abelian groups, in particular cyclic groups in Section 3. We give proofs of main theorems and some remarks in Section 4. Finally, we mention some future works related to some non-vanishing properties of $m_{\lambda}(\zeta_{(n,k)})$ and the numbers of terms of the group determinant.

First we mention the dual Cauchy kernel formula [10, Chapter I (4.2^′)]. For any partitions $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{N})$ and $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{N})$, let $\lambda\subseteq\mu$ be the inclusion partial order defined by

$$\lambda\subseteq\mu\quad\Longleftrightarrow\quad\lambda_{i}\leq\mu_{i},\quad i=1,2,\ldots,N.$$

For positive integers $M$ and $N$, the following identity holds:

$\displaystyle\prod_{i=1}^{M}\prod_{j=1}^{N}(1+x_{i}y_{j})=\sum_{\lambda\subseteq(M^{N})}e_{\lambda}(x)m_{\lambda}(y),$

where

$$e_{\lambda}(x):=\prod_{i=1}^{M}e_{\lambda_{i}}(x).$$

As a corollary of this famous result, we obtain a generating function of $m_{\lambda}(\zeta_{(n,k)})$ immediately.

From this generating function, we have integer and vanishing properties of $m_{\lambda}(\zeta_{(n,k)})$ (see Theorem 1.2 (5), (6) and Remark 4.1).

To prove Theorem 1.1 (1), we use the following lemma.

We derive a relation between $m_{\lambda}(\zeta_{(n,k)})$ and the group determinant of a cyclic group. Using the relation, we give some properties of $m_{\lambda}(\zeta_{(n,k)})$.

For a finite group $G=\{g_{1},g_{2},\ldots,g_{n}\}$, let $x_{g}$ be an indeterminate for each $g\in G$, and let $\mathbb{Z}[x_{g}]:=\mathbb{Z}[x_{g_{1}},x_{g_{2}},\ldots,x_{g_{n}}]$ be the multivariate polynomial ring in $x_{g}$ over $\mathbb{Z}$. The group determinant $\Theta(G)$ of $G$ is defined as follows (see e.g., [7], [8]):

$$\Theta(G):=\sum_{\sigma\in\mathfrak{S}_{n}}\operatorname{sgn}(\sigma)x_{g_{1}g_{\sigma(1)}^{-1}}x_{g_{2}g_{\sigma(2)}^{-1}}\cdots x_{g_{n}g_{\sigma(n)}^{-1}}\in\mathbb{Z}[x_{g}].$$

From this definition, it is evident that $\Theta(G)$ is a homogeneous polynomial of degree $n$ in $n$ variables. Furthermore, when $G$ is abelian, for any term $x_{a_{1}}x_{a_{2}}\cdots x_{a_{n}}$ in $\Theta(G)$, the product of $a_{i}$ becomes the unit element of $G$ (actually, this is true even when $G$ is non-abelian if the product is properly ordered. For a detailed explanation, see [11, Lemma 1]). From the above, for a cyclic group $\mathbb{Z}/n\mathbb{Z}=\{1,2,\ldots,n\}$, each term in $\Theta(\mathbb{Z}/n\mathbb{Z})$ is of the form $x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}$ with $n\mid i_{1}+i_{2}+\cdots+i_{n}$. Therefore, let $x_{\lambda}:=x_{\lambda_{1}}x_{\lambda_{2}}\cdots x_{\lambda_{n}}$ for $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\Lambda_{n}^{1}$, then there exist integers $c_{\lambda}$ such that

$\displaystyle\Theta(\mathbb{Z}/n\mathbb{Z})=\sum_{\begin{subarray}{c}\lambda\in\Lambda_{n}^{1}\\ |\lambda|\equiv 0\pmod{n}\end{subarray}}c_{\lambda}x_{\lambda}.$

This is one of the simplest representations of $\Theta(\mathbb{Z}/n\mathbb{Z})$ that summates similar terms.

More generally, there exist integers $c_{\lambda}$ such that

$\displaystyle\Theta\left(\mathbb{Z}/n\mathbb{Z}\right)^{k}=\sum_{\begin{subarray}{c}\lambda\in\Lambda_{n}^{k}\\ |\lambda|\equiv 0\pmod{n}\end{subarray}}c_{\lambda}x_{\lambda}.$

The following theorem implies that the coefficient $c_{\lambda}$ is equal to $m_{\lambda}(\zeta_{(n,k)})$.

To prove Theorem 3.2, we use Dedekind’s theorem. Supposing $G$ in the following is abelian, let $\widehat{G}$ be a complete set of representatives of the equivalence classes of irreducible representations for $G$ over $\mathbb{C}$. Dedekind’s theorem is as follows (see e.g., [1], [5], [14]): The group determinant $\Theta(G)$ can be factorized into irreducible polynomials over $\mathbb{C}$ as

$$\Theta(G)=\prod_{\chi\in\widehat{G}}\sum_{g\in G}\chi(g)x_{g}.$$

Mansfield [11] obtained the following lemma. Unfortunately, there is a mistake in the last sentence of the proof of [11, Lemma 3]. It says that the coefficient of $x_{e}^{n-3}x_{a}x_{b}x_{c}$ is $n$ or $2n$, but the coefficient is $n/3$ when $a=b=c$. This mistake was corrected in [15].

Lemma 3.4 will be used to prove of Theorem 1.2.

The proof of Theorem 1.1 is as follows.