Wreath Product Generalizations of the Triple $(S_{2n},H_{n},\phi)$ and Their Spherical Functions

TL;DR

This paper generalizes the Gelfand triple involving symmetric and hyperoctahedral groups to wreath products, exploring their spherical functions and connections to symmetric functions, revealing new algebraic structures and relationships.

Contribution

It introduces a wreath product generalization of the Gelfand triple and analyzes the associated spherical functions and their link to symmetric functions.

Findings

The generalized triplet remains a Gelfand triple.

Spherical functions relate to multi-partition symmetric functions.

Provides a framework for studying wreath product symmetries.

Abstract

The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $\phi$ of $H_{n}$. Their $\phi$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.

Equations

$${\nu }_2^{\xi }(\chi )=\frac{1}{|G|}\sum _{g\in G}\overline{\xi (g)}\chi (g^2)=-1,0,1(\mathrm{Theorem}\text{ref}modFS).$$

$$\rho {\sim }_{\eta }\sigma \iff {\chi }_{\rho }={\chi }_{\sigma }\mathrm{or}{\chi }_{\rho }=\overline{{\chi }_{\sigma }}\otimes \eta (\rho ,\sigma \in G^{\ast }).$$

$$\{f:G\to \mathbb{C}\mid f(gh)=f(hg)=\overline{\xi (h)}f(g)(\forall g\in G,\forall h\in H)\}.$$

$$h{v}_0=\xi (h)v_0\mathrm{and}⟨v_0,v_0{⟩}_{\gamma }=1.$$

$${\omega }_{\gamma }(g)=⟨v_0,g{v}_0{⟩}_W.$$

$${\omega }_{\gamma }=\frac{|G|}{\dim \gamma }{e}_{\gamma }{e}_{\xi },$$

$$H_n\cong ⟨(2i-1,2i),(2j-1,2j+1)(2j,2j+2)\mid 1\le i\le n,1\le j\le n-1⟩.$$

$${\varphi }_n((i,i+1))=(2i-1,2i+1)(2i,2i+2).$$

$$H{G}_n=\{(g_1,g_1,g_2,g_2,\cdots ,g_n,g_n:\sigma )\mid \sigma \in H_n\}.$$

$$(g_1,\cdots ,g_n:\sigma )v_1\otimes \cdots \otimes v_n\otimes w=g_1{v}_{{\sigma }^{-1}(1)}\otimes \cdots \otimes g_n{v}_{{\sigma }^{-1}(n)}\otimes \sigma w,$$

$$x(R_i)=\{\begin{array}{ll}(\underset{2{\rho }_1(R_i)}{\underbrace{1,\cdots 1,g_{R_i}}},\underset{2{\rho }_2(R_i)}{\underbrace{1,\cdots 1,g_{R_i}}},\cdots ,\underset{2{\rho }_{\ell (\rho (R_i))}(R_i)}{\underbrace{1,\cdots 1,g_{R_i}}},:[2\rho (R_i)]),& \rho (R_i)\ne \varnothing \\ \varnothing ,& \rho (R_i)=\varnothing .\end{array}$$

$$x(\underline{\rho })=[x(R_1),\cdots ,x(R_s){]}_{2n_1,\cdots ,2n_s}.$$

$$[\varnothing ,(e,a:(12)),(e,e,e,a^2,e,a^2:(1234)(56)){]}_{0,2,6}=(\underset{2}{\underbrace{e,a}},\underset{4,2}{\underbrace{e,e,e,a^2,e,a^2}}:(12)(3456)(78)).$$

$$\frac{1}{|G|}\sum _{g\in G}\chi (g^2)=1,-1\mathrm{or}0.$$

$${\nu }_2^{\xi }(\chi )=\frac{1}{|G|}\sum _{g\in G}\overline{\xi (g)}\chi (g^2).$$

$$\{(1,g_R;1)|R\in G_{\ast \ast }\},$$

$$(\xi \otimes \epsilon )(g,g;\sigma )=\xi (g)\epsilon (\sigma ),$$

$$UR(x)U^{-1}=\overline{R}(x),$$

$$\{\begin{array}{ll}U={}^{t}U& (\text{R is a type I}),\\ U=-{}^{t}U& (\text{R is a type II}).\end{array}$$

$$\frac{1}{|G|}\sum _{x\in G}\overline{\xi (x)}\chi (x^2)=\{\begin{array}{ll}\alpha & (\chi =\overline{\chi }\otimes \xi )\\ 0& (\chi \ne \overline{\chi }\otimes \xi ),\end{array}$$

$$\chi =\overline{\chi }\otimes \xi (\iff \overline{\chi }=\chi \otimes \overline{\xi }).$$

$$R(x)=P_{\xi ,R}{D}_{\xi }(x)\overline{R(x)}{P}_{\xi ,R}^{\ast },$$

$$\displaystyle\overline{R(x)}=\overline{P_{\xi,R}}\overline{D_{\xi}(x)}R(x){}\overline{P_{\xi,R}^{*}}.\$$

$$R(x)=P_{\xi ,R}\overline{P_{\xi ,R}}R(x)\overline{P_{\xi ,R}^{\ast }}{P}_{\xi ,R}^{\ast }.$$

$$P_{\xi ,R}\overline{P_{\xi ,R}}=\alpha E,$$

$$\frac{1}{|G|}\sum _{x\in G}\overline{\xi (x)}\chi (x^2)$$

$$=\frac{1}{|G|}\sum _{x\in G}\overline{\xi (x)}\mathrm{tr}R(x)R(x^{-1}{)}^{\ast }=\frac{1}{|G|}\sum _{x\in G}\left(\sum _{ij}\overline{\xi (x)}{r}_{ij}(x)\overline{r_{ij}(x^{-1})}\right)$$

$$=\frac{1}{|G|}\sum _{ij}\left(\sum _{x\in G}\overline{\xi (x)}{r}_{ij}(x)\overline{r_{ij}(x^{-1})}\right)=\frac{1}{|G|}\sum _{ij}\left(\sum _{x\in G}(\sum _{k,\ell }{p}_{ik}\overline{r_{k\ell }(x)}\overline{p_{j\ell }})\overline{r_{ij}(x^{-1})}\right)$$

$$=\sum _{k,\ell }\sum _{ij}{p}_{ik}\overline{p_{j\ell }}\left(\frac{1}{|G|}\sum _{x\in G}\overline{r_{k\ell }(x)}\overline{r_{ij}(x^{-1})}\right)=\frac{\mathrm{tr}{P}_{\xi ,R}\overline{P_{\xi ,R}}}{d}=\alpha .$$

$$\chi \ne \overline{\chi }\otimes \xi .$$

$$\frac{1}{|G|}\sum _{x\in G}\overline{\xi (x)}\chi (x^2)$$

$$=\frac{1}{|G|}\sum _{x\in G}\overline{\xi (x)}\mathrm{tr}R(x)R(x^{-1}{)}^{\ast }=\frac{1}{|G|}\sum _{x\in G}\left(\sum _{ij}\overline{\xi (x)}{r}_{ij}(x)\overline{r_{ij}(x^{-1})}\right)$$

Full text

000Classification number :20C05,05E05,05E10,05E35.

1. introduction

This paper deals with spherical harmonics on a finite Gelfand triple which is a generalization of the triple $(S_{2n},H_{n},\varphi)$, where $S_{2n}$ is the symmetric group of degree $2n$, $H_{n}$ is the hyperoctahedral group and $\varphi$ is an arbitrary linear character of $H_{n}$. We adopt the terminology ”Gelfand triple” by the usage of Bump’s book [2]. One of main purposes of this paper is to describe an explicit form of the irreducible decomposition of their induced representations. Furthermore we attempt to express their spherical functions as explicit as possible.

Roughly speaking, representation theory of wreath products can be said the multi-partition version of representation theory of the symmetric groups. Their irreducible representations and characters can be explicitly computed. It seems to be no new representational theoretical property of wreath products. Nevertheless, from the view point of spherical harmonics, they give us interesting mathematical objects like a multivariable version of hypergeometric functions as their zonal spherical functions [8, 9]. Furthemore, recent study of finite spherical harmonics related to wreath products succeed in finding new orthogonal polynomials and combinatorial or statistical interpretations of permutation representations [3, 10, 11]. Therefore it is important to find new finite Gelfand triples and analyze their structures as finite homogeneous spaces.

As mentioned above, the triple $(S_{2n},H_{n},\varphi)$ is a Gelfand triple, i.e. $\varphi\uparrow_{H_{n}}^{S_{2n}}$ is a multiplicity free as $S_{2n}$-module. The characters of Hecke algebra associate with $(S_{2n},H_{n},\varphi)$ are called $\varphi$-spherical functions. It is known that $\varphi$-spherical functions of this case are deeply related to zonal polynomials and Schur’s $Q$-functions [5, 12]. To be more explicit, $\varphi$-spherical functions appear in coefficients of the power sums in these symmetric functions. This fact gives an representation theoretical meaning to these symmetric functions. Algebraically there exist isomorphisms called characteristic maps between the direct sum of their Hecke algebras and the ring of symmetric functions. In this paper, we consider an generalization of the triplet by changing $S_{2n}$ into a wreath product $SG_{2n}=G\wr S_{2n}$. We define a subgroup $HG_{n}$ of $SG_{2n}$ which is a natural generalization of $H_{n}$ (details are in Section 2). In [10], the author shows the triplet $(SG_{2n},HG_{n},1)$ is Gelfand triple. When the Frobenius-Schur indicator of all irreducible characters of $G$ is equal to [math] or 1, their spherical functions are obtained as the coefficients of multi-partition version of power sum in product of Schur functions and zonal polynomials. Here we have to explain multi-partition version of symmetric functions. For a finite set $A$, we prepare the power sums $p_{r}(a)$ $(a\in A)$ and set $\Lambda[A]={\mathbb{C}}\langle p_{r}(a)\mid a\in A,r\geq 1\rangle$. We call $\Lambda[A]$ the multi-partition version of the ring of symmetric functions associated with $A$. In the book [7], the character theory of wreath products is introduced in terms of $\Lambda[G^{*}]$, where $G^{*}$ denotes the set of the irreducible representations of a finite group $G$. Recently Ingram, Jing and Stitzinger study this type of symmetric functions in [13].

One of our main result of this paper is to show that a triplet $(SG_{2n},HG_{n},\Theta_{\xi,\pi})$ is a Gelfand triple for an arbitrary linear character $\Theta_{\xi,\pi}$ of $HG_{n}$. Here all linear representations of $HG_{n}$ can be indexed by linear representations $\xi$ of $G$ and $\pi$ of $H_{n}$ (see. Section 4.1). The irreducible representations of $SG_{2n}$ are indexed by a $|G^{*}|$-tuple of partitions $(\lambda(\chi)\mid\chi\in G^{*})$. The problem we should consider is: which partitions appear as irreducible component of $\Theta_{\xi,\pi}\uparrow_{HG_{n}}^{SG_{2n}}$? To answer this problem, we need to show a twisted version Frobenius-Schur theorem:

[TABLE]

We see that these three values 1, 0 and -1 determine the type of a partition $\lambda(\chi)$ in a irreducible component indexed by $(\lambda(\chi)\mid\chi\in G^{*})$ of the induced representation. After completing this problem, we consider the $\Theta_{\xi,\pi}$-spherical functions of $(SG_{2n},HG_{n},\Theta_{\xi,\pi})$. We are greatly interested in $\Theta_{\xi,\pi}$-spherical function because they gives sometimes important examples of orthogonal functions. Actually our $\Theta_{\xi,\pi}$-spherical functions appear as coefficients of multi-power sum of product including Schur functions, Schur’s $Q$-functions and Jack polynomials at the parameter $\alpha=2$ and $1/2$.

This paper is organized as follows. We prepare the almost all notations used in this paper in Section 2. In section 3, we consider a twisted version of the Frobenius-Schur formula through an analysis of $(SG_{2},HG_{1},\Theta_{\xi,\pi})$. As conclusion of this section, we have a certain relation between the number of irreducible representations of finite groups and that of conjugacy classes. Section 4 is devoted to description of the irreducible decomposition of our triplet. Simultaneously we discuss basis of the Hecke algebra. The main tool of this section is the relation obtained in Section 3. In section 5, we consider spherical functions. Two special cases are computed explicitly. In section 6, we obtain the relation between $\Theta_{\xi,\pi}$-spherical functions and multi-partition version of symmetric functions. We construct an graded algebra which are the direct sum of Hecke algebra. We make an isomorphism $CH_{\pi}$ between the graded Hecke algebra and a certain multi-partition version of the symmetric functions. We see that the image of $\Theta_{\xi,\pi}$-spherical functions under $CH_{\pi}$ is products of some symmetric functions. In the last section, we apply the spherical harmonics of our triplet to the case of $G={\mathbb{Z}}/2{\mathbb{Z}}$ and see a their $\Theta_{\xi,\pi}$-spherical functions are essentially same as the characters of symmetric groups.

2. Preliminaries

Let $G$ be a finite group. We denote by $G_{*}$ (resp. $G^{*}$) the set of the conjugacy classes (resp. a complete representatives of the irreducible representations) of $G$. Let ${\mathcal{L}}(G)$ be the set of all linear characters of $G$. For $g\in G$, $C_{g}$ denotes the conjugacy class of $g$. Put $\zeta_{g}=\frac{|G|}{|C_{g}|}$. For $\gamma\in G^{*}$, we denote by $\chi_{\gamma}$ the character of a representation $\gamma$. We always identify $\{\chi_{\gamma}|\gamma\in G^{*}\}$ with $G^{*}$. Fix $\eta\in{\mathcal{L}}(G)$. Put $G_{**}=\{C_{g}\cup C_{g^{-1}}|g\in G\}$ and $G^{**}_{\eta}=\{\rho|\rho\in G_{*}\}/_{\sim_{\eta}}$, where,

[TABLE]

By fixing a representatives, we identify $G_{\eta}^{**}$ with a subset of $G^{*}$. Put $n_{**}=|G_{**}|$ and $n_{**}^{\eta}=|G^{**}_{\eta}|$. We remark $n^{**}=n_{**}^{1}$, where 1 is the trivial representation of $G$. If $C_{g}=C_{g^{-1}}$ (resp. $C_{g}\not=C_{g^{-1}}$), then we call $C_{g}$ and $R_{g}=C_{g}\cup C_{g^{-1}}$ a real (resp. a complex).

Example 2.1**.**

In the case of $G=C_{6}=\langle a\mid a^{6}=1\rangle$ , we set $\xi_{m}:a\mapsto\exp\frac{2m\pi i}{6}\ (0\leq m\leq 5)$. Then we have $G_{**}=\{\{1\},\{a,a^{5}\},\{a^{2},a^{4}\},\{a^{3}\}\}$ and $G_{\xi_{1}}^{**}=\left\{\{\xi_{0},\xi_{1}\},\{\xi_{2},\xi_{5}\},\{\xi_{3},\xi_{4}\}\right\}$.

Denote by $\mathbb{C}G$ the group algebra of $G$. We identify $f=\sum_{x\in G}f(x)x\in\mathbb{C}G$ with the function $x\mapsto f(x).$ Under this identification, the multiplication in $\mathbb{C}G$ is given by the convolution: $(f_{1}f_{2})(x)=\sum_{y\in G}f_{1}(y^{-1})f_{2}(yx)$. Let $H$ be a subgroup of $G$. For $\xi\in{\mathcal{L}}(H)$, let $e_{\xi}=\frac{1}{|H|}\sum_{h\in H}\xi(h^{-1})h$. A subalgebra ${\mathcal{H}}^{\xi}(G,H)=e_{\xi}{\mathbb{C}}Ge_{\xi}$ of $\mathbb{C}G$ is called a Hecke algebra of a triplet $(G,H,\xi)$. The Hecke algebra ${\mathcal{H}}^{\xi}(G,H)$ can be identified with

[TABLE]

We denote by $\xi\uparrow_{H}^{G}$ a representation of $G$ induced from $\xi$. If $\xi\uparrow^{G}_{H}$ is multiplicity-free as $G$-module, then we call the triplet $(G,H,\xi)$ a Gelfand triple. From Schur’s lemma, “ $\xi\uparrow^{G}_{H}$ is multiplicity-free” and “$\mathcal{H}^{\xi}(G,H)$ is commutative” are equivalent. In this paper, we only consider representations of finite groups. Therefore we assume that each representation space $V$ of $G$ has $G$-invariant inner product $\langle,\rangle_{V}$. We assume that $(G,H,\xi)$ is a Gelfand triple. Let $\gamma\in G^{*}$ be an irreducible component of $\xi\uparrow_{H}^{G}$. Then there is a unique element of $\gamma$ satisfying

[TABLE]

We define a $\xi$-spherical function by

[TABLE]

Also $\xi$-spherical function has the following expression;

[TABLE]

where $e_{\gamma}=\frac{\dim\gamma}{|G|}\sum_{g\in G}\chi_{\gamma}(g^{-1})g$.

Let $\lambda$ be a partition and let $P_{n}$ be the set of all partitions of $n$. We denote by $\lambda^{\prime}$ the transpose of $\lambda$. For $\lambda$, $m_{i}(\lambda)$ denotes the multiplicity of $i$ in $\lambda$. Let $\ell(\lambda)$ be the length of $\lambda$. If all parts of $\lambda$ are even (resp. odd), then we call it a even (resp. odd) partition. We denote by $EP_{n}$ (resp. $OP_{n}$) the set of all even (resp. odd) partitions of $n$. Let $SP_{n}$ be the set of all strict partitions of $n$. Set $P=\cup_{n\geq 0}P_{n}$, $SP=\cup_{n\geq 0}SP_{n}$, $OP=\cup_{n\geq 0}OP_{n}$ and $EP=\cup_{n\geq 0}EP_{n}$. We denote by $D(\mu)$ the doubling of $\mu\in SP$ (cf. [12]). For example, $D(421)=(5441)$. Let $\underline{\lambda}=(\lambda(x)|x\in X)$ be a $|X|$-tuple of partitions for a set $X$. We define the weight of $\underline{\lambda}$ by $|\underline{\lambda}|=\sum_{x\in X}|\lambda(x)|$. For two partitions $\lambda$ and $\mu$, we define a partition $\lambda\cup\mu$ by $m_{i}(\lambda\cup\mu)=m_{i}(\lambda)+m_{i}(\mu)$ for any $i$. For $\underline{\lambda}=(\lambda(x)|x\in X)$ and $\underline{\mu}=(\mu(x)|x\in X)$, set $\underline{\lambda}\cup\underline{\mu}=(\lambda(x)\cup\mu(x)|x\in X)$. For $(\rho(x)\mid x\in X)$, we define $\hat{\underline{\rho}}$ to be a partition $\bigcup_{x\in X}\rho(x)$.

The hyperoctahedral group $H_{n}$ is a subgroup of $S_{2n}$ defined by

[TABLE]

$H_{n}$ is the centralizer of an element $(12)(34)\cdots(2n-1,2n)\in S_{2n}$. Here we remark $\langle(2j-1,2j+1)(2j,2j+2)\mid 1\leq i\leq n,1\leq j\leq n-1\rangle\cong S_{n}=\langle(i,i+1)|1\leq i\leq n\!-\!1\rangle.$ We define an isomorphism $\phi_{n}$ from $S_{n}$ onto $\langle(2j-1,2j+1)(2j,2j+2)|1\leq i\leq n,1\leq j\leq n-1\rangle\subset H_{n}$ by

[TABLE]

We consider the wreath product $SG_{n}=G\wr S_{n}=\{(g_{1},g_{2},\cdots,g_{n}:\sigma)\mid g_{i}\in G,\sigma\in S_{n}\}$ of a finite group $G$ with a symmetric group $S_{n}$. We define a subgroup $HG_{n}$ of $SG_{2n}$ by

[TABLE]

Put $\underline{\lambda}=(\lambda(\chi)\mid\chi\in G^{*})$ and $|\underline{\lambda}|=n$. We denote by $S(\underline{\lambda})$ the irreducible representation of $SG_{n}$