The algebra of conjugacy classes of the wreath product
of a finite group with the symmetric group

Omar Tout Department of Mathematics, College of Science, Sultan Qaboos University, P. O Box 36, Al Khod 123, Sultanate of Oman [email protected]

Abstract.

For a finite group $G,$ we define the concept of $G$ -partial permutation and use it to show that the structure coefficients of the center of the wreath product $G\wr\mathcal{S}_{n}$ algebra are polynomials in $n$ with non-negative integer coefficients. Our main tool is a combinatorial algebra which projects onto the center of the group $G\wr\mathcal{S}_{n}$ algebra for every $n.$ This generalizes the Ivanov and Kerov method to prove the polynomiality property for the structure coefficients of the center of the symmetric group algebra.

Key words and phrases:

Wreath product, partial permutations, structure coefficients, character theory, shifted symmetric functions

2020 Mathematics Subject Classification:

Primary 05E05, 05E10, 20C30; Secondary 20E22.

1. Introduction

Throughout this paper $G$ will be a finite group, $1_{G}$ its identity element, $G_{\star}$ its set of conjugacy classes and $G^{\star}$ its set of irreducible complex characters. If $n$ is a positive integer, let $\mathcal{S}_{n}$ denote the symmetric group on the set $[n]:=\{1,2,\ldots,n\}.$ A partition is a finite list of non-increasing positive integers called parts. The size of a partition is the sum of all of its parts. A partition of size $n$ is usually called a partition of $n.$ We denote by $\mathcal{P}_{n}^{G_{\star}}$ the set of families of partitions $\Lambda=(\Lambda(c))_{c\in G_{\star}},$ indexed by $G_{\star},$ such that the sizes of the partitions $\Lambda(c)$ sum up to $n.$ The type of an element of the wreath product $G\wr\mathcal{S}_{n}$ is a family of partitions $\Lambda\in\mathcal{P}_{n}^{G_{\star}},$ see [5]. Two elements of $G\wr\mathcal{S}_{n}$ are conjugate if and only if they have the same type. The center of the group $G\wr\mathcal{S}_{n}$ algebra, which will be denoted $Z(\mathbb{C}[G\wr\mathcal{S}_{n}]),$ is the algebra over $\mathbb{C}$ generated by the conjugacy classes of $G\wr\mathcal{S}_{n}.$ If $\Lambda\in\mathcal{P}_{n}^{G_{\star}},$ we define $\mathbf{C}_{\Lambda}$ to be the formal sum of all the elements in $G\wr\mathcal{S}_{n}$ with type $\Lambda.$ The family $(\mathbf{C}_{\Lambda})_{\Lambda}$ indexed by $\mathcal{P}_{n}^{G_{\star}}$ is a linear basis for $Z(\mathbb{C}[G\wr\mathcal{S}_{n}]).$ The structure coefficients $c_{\Lambda\Delta}^{\Gamma}$ are the non-negative integers defined by the following product in $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$

\mathbf{C}_{\Lambda}\mathbf{C}_{\Delta}=\sum_{\Gamma\in\mathcal{P}_{n}^{G_{\star}}}c_{\Lambda\Delta}^{\Gamma}\mathbf{C}_{\Gamma}.

In the case where $G$ is the trivial group, the group $G\wr\mathcal{S}_{n}$ is isomorphic to the symmetric group $\mathcal{S}_{n}.$ The conjugacy classes of $\mathcal{S}_{n}$ are indexed by partitions of $n.$ It is a difficult problem to find explicit formulas even for particular structure coefficients of $Z(\mathbb{C}[\mathcal{S}_{n}]),$ see [4], [2], [11]. In [1], Farahat and Higman showed that the structure coefficients of $Z(\mathbb{C}[\mathcal{S}_{n}])$ are polynomials in $n.$ By introducing partial permutations in [3], Ivanov and Kerov gave a combinatorial proof to this result. Recently, we used in [10] our general framework developed in [9] to show a polynomiality property for the structure coefficients of $Z(\mathbb{C}[\mathcal{S}_{k}\wr\mathcal{S}_{n}]).$

Beside from being combinatorial, the Ivanov-Kerov approach, developed in [3], uses a universal algebra which turns out to be isomorphic to the algebra of shifted symmetric functions. In the past few years, it was used to show a polynomiality property for the structure coefficients of some interesting algebras. For example, we define the notion of partial bijection in [8] to show that the structure coefficients of the Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_{n}),$ where $\mathcal{B}_{n}$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n},$ are polynomials in $n.$ In [6], the concept of partial isomorphism appeared to give a polynomiality property for the structure coefficients of the center of the group $\operatorname{GL}(n,\mathbb{F}_{q})$ algebra, where $q$ is a prime number and $\operatorname{GL}(n,\mathbb{F}_{q})$ is the group of invertible $n\times n$ matrices with coefficients in the finite field $\mathbb{F}_{q}.$ We used the notion of $k$ -partial permutation in [12] to give a more combinatorial proof to our result in [10].

In [13], Wang proved that the structure coefficients $c_{\Lambda\Delta}^{\Gamma}$ of $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$ are polynomials in $n.$ He used the Farahat-Higman approach developed in [1] for the center of the symmetric group algebra. The goal of this paper is to generalize the Ivanov-Kerov approach in order to obtain Wang’s result by a more algebraic combinatorial way. For this reason, we will define the concept of $G$ -partial permutation and use it to build a universal combinatorial algebra which projects onto the center of the group $G\wr\mathcal{S}_{n}$ algebra for each $n.$ We will prove that this universal algebra is isomorphic to the algebra of shifted symmetric functions on $|G^{\star}|$ alphabets. Recently, it came to our attention that Wang mentioned our generalization in [14, Section $5.3$ ]. However, in addition to providing all the details, we think that some presented results like Theorem 7.1, are new and make a valuable contribution to the literature.

The paper is organized as follows. In Section 2, we present the necessary definitions for partitions and we review some basic results concerning the conjugacy classes and the center of the group $G\wr\mathcal{S}_{n}$ algebra. Then, in Section 3, we introduce the notion of $G$ -partial permutation. An action of the group $G\wr\mathcal{S}_{n}$ on the set of $G$ -partial permutations of $n$ is given in Section 4. The universal combinatorial algebra $\mathcal{A}_{\infty}^{G},$ which projects on the center of the group $G\wr\mathcal{S}_{n}$ algebra for each $n,$ will be built in Section 5. Next in Section 6, we prove in Theorem 6.2 that the structure coefficients of the center of the group $G\wr\mathcal{S}_{n}$ algebra are polynomials in $n.$ In the last section, we present an isomorphism between $\mathcal{A}_{\infty}^{G}$ and the algebra of shifted symmetric functions on $|G^{\star}|$ alphabets.

2. Algebra of the conjugacy classes of $G\wr\mathcal{S}_{n}$

In this section we will review all necessary definitions and results concerning the conjugacy classes of $G\wr\mathcal{S}_{n}.$ For more details, the reader is invited to check [5, Appendix B].

2.1. Partitions

A partition $\lambda$ is a weakly decreasing list of positive integers $(\lambda_{1},\ldots,\lambda_{l}).$ The $\lambda_{i}$ are called the parts of $\lambda.$ The size of $\lambda,$ denoted by $|\lambda|,$ is the sum of all of its parts. We say that $\lambda$ is a partition of $n$ if $|\lambda|=n.$ The set of all partitions of $n$ will be denoted $\mathcal{P}_{n}.$ In this paper, we will mainly use the exponential notation $\lambda=(1^{m_{1}(\lambda)},2^{m_{2}(\lambda)},3^{m_{3}(\lambda)},\ldots),$ where $m_{i}(\lambda)$ is the number of parts equal to $i$ in the partition $\lambda.$ We will dismiss $i^{m_{i}(\lambda)}$ from $\lambda$ when $m_{i}(\lambda)=0,$ for example, we will write $\lambda=(1,2,4^{2})$ instead of $\lambda=(1,2,3^{0},4^{2},5^{0},\ldots).$ If $\lambda$ and $\delta$ are two partitions, we define the union $\lambda\cup\delta$ and subtraction $\lambda\setminus\delta$ (if exists) as the following partitions:

\lambda\cup\delta=(1^{m_{1}(\lambda)+m_{1}(\delta)},2^{m_{2}(\lambda)+m_{2}(\delta)},3^{m_{3}(\lambda)+m_{3}(\delta)},\ldots).

\lambda\setminus\delta=(1^{m_{1}(\lambda)-m_{1}(\delta)},2^{m_{2}(\lambda)-m_{2}(\delta)},3^{m_{3}(\lambda)-m_{3}(\delta)},\ldots)\text{ if $m_{i}(\lambda)\geq m_{i}(\delta)$ for any $i.$ }

The cycle-type of a permutation of $\mathcal{S}_{n}$ is the partition of $n$ obtained from the lengthes of the cycles that appear in its decomposition into product of disjoint cycles. For example, the permutation $(1,4)(2,6,3)(5)(7,8)$ of $\mathcal{S}_{8}$ has cycle-type $(1,2^{2},3).$ It is well known that two permutations of $\mathcal{S}_{n}$ belong to the same conjugacy class if and only if they have the same cycle-type. Thus the conjugacy classes of $\mathcal{S}_{n}$ can be indexed by partitions of $n.$ The conjugacy class of $\mathcal{S}_{n}$ associated to the partition $\lambda=(1^{m_{1}(\lambda)},2^{m_{2}(\lambda)},3^{m_{3}(\lambda)},\ldots,n^{m_{n}(\lambda)})\in\mathcal{P}_{n}$ will be denoted $C_{\lambda}$ and its cardinal is given by the following formula:

|C_{\lambda}|=\frac{n!}{z_{\lambda}},

where

z_{\lambda}:=\prod_{i\geq 1}i^{m_{i}(\lambda)}m_{i}(\lambda)!.

A partition is called proper if it does not have any part equal to 1. The proper partition associated to a partition $\lambda$ is the partition $\bar{\lambda}:=(2^{m_{2}(\lambda)},3^{m_{3}(\lambda)},\ldots).$ The set of all proper partitions with size less than or equal to $n$ will be denoted $\mathcal{PP}_{\leq n}.$ If $\lambda$ is a partition of $r<n,$ we can extend $\lambda$ to a partition of $n$ by adding $n-r$ parts equal to one, the new partition of $n$ will be denoted $\underline{\lambda}_{n}.$

If $X$ is a finite set and $\Lambda=(\Lambda(x))_{x\in X}$ is a family of partitions indexed by $X,$ we define the size of $\Lambda,$ denoted by $|\Lambda|,$ to be the sum of the sizes of $\Lambda(x)$

|\Lambda|=\sum_{x\in X}|\Lambda(x)|.

The set of families of partitions of size $n$ indexed by $X$ will be denoted $\mathcal{P}_{n}^{X}$ and $\mathcal{P}_{\leq n}^{X}$ will denote the set of families of partitions indexed by $X$ with size less than or equal to $n.$ In this paper we will mainly encounter families of partitions indexed by $G_{\star}$ and $G^{\star}.$ An element $\Lambda\in\mathcal{P}_{n}^{G_{\star}}$ is called proper if the partition $\Lambda(\{1_{G}\})$ is proper. We will use $\mathcal{PP}_{n}^{G_{\star}}$ (resp. $\mathcal{PP}_{\leq n}^{G_{\star}}$ ) to denote the set of proper families of partitions of size $n$ (resp. less than or equal to $n$ ) indexed by $G_{\star}.$ If $\Lambda\in\mathcal{P}_{\leq n}^{G_{\star}},$ we define $\underline{\Lambda}_{n}$ to be the element of $\mathcal{P}_{n}^{G_{\star}}$ with $\underline{\Lambda}_{n}(c)=\Lambda(c)$ if $c\neq\{1_{G}\}$ and

\underline{\Lambda}_{n}(\{1_{G}\})=\Lambda(\{1_{G}\})\cup(1^{n-|\Lambda|})=\underline{\Lambda(\{1_{G}\})}_{n}.

2.2. Conjugacy classes of $G\wr\mathcal{S}_{n}$

The wreath product $G\wr\mathcal{S}_{n}$ is the group with underlying set $G^{n}\times\mathcal{S}_{n}$ and product defined as follows:

((\sigma_{1},\ldots,\sigma_{n});p)\cdot((\epsilon_{1},\ldots,\epsilon_{n});q)=((\sigma_{q^{-1}(1)}\epsilon_{1},\ldots,\sigma_{q^{-1}(1)}\epsilon_{n});pq),

for any $((\sigma_{1},\ldots,\sigma_{n});p),((\epsilon_{1},\ldots,\epsilon_{n});q)\in G^{n}\times\mathcal{S}_{n}.$ We apply $p$ before $q$ when we write the product $pq.$ The identity in this group is $(1;1):=((1_{G},1_{G},\ldots,1_{G});\operatorname{Id}_{n}).$ The inverse of an element $((\sigma_{1},\sigma_{2},\ldots,\sigma_{n});p)\in G\wr\mathcal{S}_{n}$ is given by

((\sigma_{1},\sigma_{2},\ldots,\sigma_{n});p)^{-1}=((\sigma^{-1}_{p(1)},\sigma^{-1}_{p(2)},\ldots,\sigma^{-1}_{p(n)});p^{-1}).

Let $x=(g;p)\in G\wr\mathcal{S}_{n},$ where $g=(g_{1},\ldots,g_{n})\in G^{n}$ and $p\in\mathcal{S}_{n}$ is written as a product of disjoint cycles. If $(i_{1},i_{2},\ldots,i_{r})$ is a cycle of $p,$ the element $g_{i_{r}}g_{i_{r-1}}\ldots g_{i_{1}}\in G$ is determined up to conjugacy in $G$ by $g$ and $(i_{1},i_{2},\ldots,i_{r}),$ and is called the cycle product of $x$ corresponding to the cycle $(i_{1},i_{2},\ldots,i_{r}),$ see [5, Page $170$ ]. For any conjugacy class $c\in G_{\star},$ we denote by $\rho(c)$ the partition written in the exponential way where $m_{i}(\rho(c))$ is the number of cycles of length $i$ in $p$ whose cycle-product lies in $c$ for each integer $i\geq 1.$ Then each element $x=(g;p)\in G\wr\mathcal{S}_{n}$ gives rise to a family of partitions $(\rho(c))_{c\in G_{\star}}$ indexed by $G_{\star}$ such that

\sum_{i\geq 1,c\in G_{\star}}im_{i}(\rho(c))=n.

This family of partitions is called the type of $x$ and denoted $\operatorname{type}(x).$

Example 2.1.

When $G=\mathbb{Z}_{k},$ the type of $x=(g;p)\in\mathbb{Z}_{k}\wr\mathcal{S}_{n}$ is a $k$ -vector of partitions $\Lambda=(\lambda_{0},\lambda_{1},\ldots,\lambda_{k-1})$ where each partition $\lambda_{i}$ is formed out of cycles $c$ of $p$ whose cycle product equals $i.$ For example, consider the element $x=(g,p)\in\mathbb{Z}_{3}\wr\mathcal{S}_{10}$ where $g=(1,0,2,0,0,1,1,2,1,0)$ and $p=(1,4)(2,5)(3)(6)(7,8,9,10).$ The cycle product of $(1,4)$ is $1+0=1,$ of $(2,5)$ is $0+0=0,$ of $(3)$ is $2,$ of $(6)$ is $1$ and of $(7,8,9,10)$ is $1+2+1+0=1$ in $\mathbb{Z}_{3}.$ Thus $\operatorname{type}(x)=(\lambda_{0},\lambda_{1},\lambda_{2})$ with $\lambda_{0}=(2),$ $\lambda_{1}=(4,2,1)$ and $\lambda_{2}=(1).$

It turns out, see [5, Page $170$ ], that two permutations are conjugate in $G\wr\mathcal{S}_{n}$ if and only if they have the same type. Thus the conjugacy classes of $G\wr\mathcal{S}_{n}$ can be indexed by the elements of $\mathcal{P}_{n}^{G_{\star}}.$ If $\Lambda\in\mathcal{P}_{n}^{G_{\star}},$ we will denote by $C_{\Lambda}$ its associated conjugacy class:

C_{\Lambda}:=\{x\in G\wr\mathcal{S}_{n};\operatorname{type}(x)=\Lambda\}.

From [5, (3.1)], the order of the centralizer of an element of type $\Lambda$ in $G\wr\mathcal{S}_{n}$ is

Z_{\Lambda}=\prod_{c\in G_{\star}}z_{\Lambda(c)}\xi_{c}^{l(\Lambda(c))},

where $\xi_{c}=\frac{|G|}{|c|}$ is the order of the centralizer of an element $g\in c$ in $G.$ Thus, if $\Lambda\in\mathcal{P}_{n}^{G_{\star}},$ the cardinal of $C_{\Lambda}$ is given by:

|C_{\Lambda}|=\frac{|G\wr\mathcal{S}_{n}|}{Z_{\Lambda}}=\frac{|G|^{n}n!}{\prod\limits_{c\in G_{\star}}z_{\Lambda(c)}\xi_{c}^{l(\Lambda(c))}}.

2.3. The center of the group $G\wr\mathcal{S}_{n}$ algebra

The group algebra of $G\wr\mathcal{S}_{n},$ denoted by $\mathbb{C}[G\wr\mathcal{S}_{n}],$ is the algebra over $\mathbb{C}$ with basis the elements of the group $G\wr\mathcal{S}_{n}.$ The product in $\mathbb{C}[G\wr\mathcal{S}_{n}]$ is the linear extension of the group product in $G\wr\mathcal{S}_{n}.$ The center of the group algebra $\mathbb{C}[G\wr\mathcal{S}_{n}],$ usually denoted by $Z(\mathbb{C}[G\wr\mathcal{S}_{n}]),$ is the sub-algebra of $\mathbb{C}[G\wr\mathcal{S}_{n}]$ of invariant elements under the conjugation action of $G\wr\mathcal{S}_{n}$ on $\mathbb{C}[G\wr\mathcal{S}_{n}]:$

Z(\mathbb{C}[G\wr\mathcal{S}_{n}]):=\{x\in\mathbb{C}[G\wr\mathcal{S}_{n}];yx=xy~{}~{}~{}~{}\forall y\in G\wr\mathcal{S}_{n}\}.

The conjugacy classes of $G\wr\mathcal{S}_{n}$ index a basis of $Z(\mathbb{C}[G\wr\mathcal{S}_{n}]).$ We showed in Section 2.2 that the conjugacy classes of $G\wr\mathcal{S}_{n}$ are indexed by the elements of $\mathcal{P}_{n}^{G_{\star}}.$ Thus, the family $({\bf C}_{\Lambda})_{\Lambda\in\mathcal{P}_{n}^{G_{\star}}},$ where

{\bf C}_{\Lambda}=\sum_{x\in C_{\Lambda}}x,

forms a linear basis for $Z(\mathbb{C}[G\wr\mathcal{S}_{n}]).$ Let $\Lambda$ and $\Delta$ be two elements of $\mathcal{P}_{n}^{G_{\star}},$ the structure coefficients $c_{\Lambda\Delta}^{\Gamma}$ of the algebra $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$ are defined by the following equation:

(1)

\mathbf{C}_{\Lambda}\mathbf{C}_{\Delta}=\sum_{\Gamma\in\mathcal{P}_{n}^{G_{\star}}}c_{\Lambda\Delta}^{\Gamma}\mathbf{C}_{\Gamma}.

The coefficients $c_{\Lambda\Delta}^{\Gamma}$ are non-negative integer since they count the number of pairs of elements $(x,y)\in C_{\Lambda}\times C_{\Delta}$ such that $x\cdot y=z$ for a fixed element $z\in C_{\Gamma}.$ However, it is a very hard problem to compute these coefficients even in particular cases. For instance, the easiest choice for $G$ is the trivial group in which the group $G\wr\mathcal{S}_{n}$ is isomorphic to $\mathcal{S}_{n}.$ There is no explicit formula to compute all the structure coefficients of the center of the symmetric group algebra. Explicit formulas for particular structure coefficients of $Z(\mathbb{C}[\mathcal{S}_{n}])$ appeared in many papers, for example see [4], [2] and [11].

In [1], Farahat and Higman showed that the structure coefficients of $Z(\mathbb{C}[\mathcal{S}_{n}])$ are polynomials in $n$ which was later proved by Ivanov and Kerov in [3] using a more combinatorial way. In [13], following the Farahat and Higman approach, Wang proved that the structure coefficients $c_{\Lambda\Delta}^{\Gamma}$ of $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$ are polynomials in $n.$ In the next sections we will develop a combinatorial approach in order to prove Wang’s result using the Ivanov-Kerov method.

3. $G$ -partial permutations

A partial permutation of $[n]$ is a pair $(d,\omega)$ consisting of an arbitrary subset $d$ of $[n]$ and an arbitrary bijection $\omega:d\longrightarrow d.$ The notion of partial permutation of $[n]$ appeared in [3] to show by a combinatorial way that the structure coefficients of the center of the symmetric group $\mathcal{S}_{n}$ algebra are polynomials in $n.$

If $d$ is a subset of $[n],$ we denote by $G^{n}_{d}$ the set of vectors $g$ with $n$ -coordinates such that $g_{i}\in G$ if $i\in d$ and $g_{i}$ is left blank otherwise. For example, if $G=\mathbb{Z}_{3},$ $n=5$ and $d=\{1,3,4\}$ then $(1,,2,0,)\in G^{5}_{d}$ but $(1,,1,1,1)\notin G^{5}_{d}.$

Definition 3.1.

A $G$ -partial permutation of $[n]$ is a pair $(g;(d,\omega))$ where $(d,\omega)$ is a partial permutation of $[n]$ and $g\in G^{n}_{d}.$

We denote by $\mathfrak{P}^{G}_{n}$ the set of all $G$ -partial permutations of $[n].$ It would be clear that

|\mathfrak{P}^{G}_{n}|=\sum_{k=0}^{n}{n\choose k}k!|G|^{k}.

A $G$ -partial permutation $(g;(d,\omega))$ of $[n]$ may be represented by a diagram obtained by drawing the two lines permutation diagram associated to $(d,\omega),$ with the nodes of the bottom row replaced by the elements $g_{i}$ for $i\in d.$ This representation will help us understanding the product between $G$ -partial permutations of $[n]$ which will be defined later.

Example 3.2.

If $n=9,$ $A=\{2,4,5,6\}$ and $\omega=(2,5)(4,6),$ then we represent the element $(g;(A,\omega))$ by the following diagram

The definition of type can be extended naturally to a $G$ -partial permutation of $[n].$ If $x=(g;(d,\omega))$ is a $G$ -partial permutation of $[n]$ and $c\in G_{\star}$ then let $\rho(c)$ be the partition written in the exponential way where $m_{i}(\rho(c))$ is the number of cycles of length $i$ in $\omega$ whose cycle-product lies in $c$ for each integer $i\geq 1.$ Define the type of $x$ to be the family of partitions $(\rho(c))_{c\in G_{\star}}$ indexed by $G_{\star}.$ It would be clear that

|\operatorname{type}(x)|=|d|.

If $x=(g;(d,\omega))\in\mathfrak{P}^{G}_{n},$ we denote by $\widetilde{x}$ the element $(\widetilde{g};\widetilde{\omega})$ of $G\wr\mathcal{S}_{n}$ where $\widetilde{\omega}$ and $\widetilde{g}$ are defined by:

\widetilde{\omega}(a)=\left\{\begin{array}[]{ll}\omega(a)&\qquad\mathrm{if}\quad a\in d,\\ a&\qquad\mathrm{if}\quad a\in[n]\setminus d.\\ \end{array}\right.\text{ and }\widetilde{g}_{i}=\left\{\begin{array}[]{ll}g_{i}&\qquad\mathrm{if}\quad i\in d,\\ 1_{G}&\qquad\mathrm{if}\quad i\in[n]\setminus d.\\ \end{array}\right.

The product of two $G$ -partial permutations $(g;(d_{1},\omega_{1}))$ and $(h;(d_{2},\omega_{2}))$ of $[n]$ is defined by:

\big{(}g;(d_{1},\omega_{1})\big{)}\cdot\big{(}h;(d_{2},\omega_{2})\big{)}=\big{(}(\widetilde{g}_{\widetilde{\omega}_{2_{|d_{1}\cup d_{2}}}^{-1}(1)}\widetilde{h}_{1},\ldots,\widetilde{g}_{\widetilde{\omega}_{2_{|d_{1}\cup d_{2}}}^{-1}(n)}\widetilde{h}_{n});(d_{1},\omega_{1})(d_{2},\omega_{2})\big{)},

where

(d_{1},\omega_{1})(d_{2},\omega_{2})=(d_{1}\cup d_{2},\widetilde{\omega}_{1_{|d_{1}\cup d_{2}}}\widetilde{\omega}_{2_{|d_{1}\cup d_{2}}}).

This product is well defined since $(d_{1},\omega_{1})(d_{2},\omega_{2})$ is a partial permutation of $[n].$ The set $\mathfrak{P}^{G}_{n}$ is a semigroup with this multiplication. The unity in $\mathfrak{P}^{G}_{n}$ is the $G$ -partial permutation $((,,\ldots,);(\emptyset,e_{0}))$ where $e_{0}$ is the trivial permutation of the empty set $\emptyset.$ We denote by $\mathcal{B}^{G}_{n}=\mathbb{C}[\mathfrak{P}^{G}_{n}]$ the algebra of the semigroup $\mathfrak{P}^{G}_{n}.$

Example 3.3.

Reconsider the $G$ -partial permutation $(g;(A,\omega))$ of Example 3.2 and let $(f;(B,\sigma))$ be the $G$ -partial permutation of $[9]$ with $B=\{1,3,5,6,8,9\}$ and $\sigma=(1,5,8)(3,9)(6)$ then the product $(g;(A,\omega))\cdot(f;(B,\sigma))$ yields the following $G$ -partial permutation of $[9]$

\big{(}(f_{1},g_{2},f_{3},g_{4},f_{5},g_{6}f_{6},,g_{5}f_{8},f_{9});(\{1,2,3,4,5,6,8,9\},(1,5,2,8)(3,9)(4,6))\big{)}

This can be obtained easily by drawing the diagram of $(g;(A,\omega))$ above the diagram of $(f;(B,\sigma))$ then extending both of them to $A\cup B$ as represented below

All extensions are drawn in red. The diagram of the product $(g;(A,\omega))\cdot(f;(B,\sigma))$ is then obtained by taking the resulted diagram of the above combination

4. Action of $G\wr\mathcal{S}_{n}$ on $\mathfrak{P}^{G}_{n}$

Any element $(g;\sigma)$ of the wreath product $G\wr\mathcal{S}_{n}$ can be seen as a $G$ -partial permutation of $[n]$ by identifying $\sigma$ with the partial permutation $([n],\sigma).$ The wreath product $G\wr\mathcal{S}_{n}$ acts on the semigroup $\mathfrak{P}^{G}_{n}$ by:

(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}:=\big{(}f;(\sigma^{-1}(d),\sigma\omega\sigma^{-1})\big{)},

where $f_{i}=g_{\omega^{-1}(\sigma(i))}h_{\sigma(i)}g^{-1}_{\sigma(i)}$ if $i\in\sigma^{-1}(d)$ for any $(g;\sigma)\in G\wr\mathcal{S}_{n}$ and $\big{(}h;(d,\omega)\big{)}\in\mathfrak{P}^{G}_{n}.$ The orbits of this action will be called the conjugacy classes of $\mathfrak{P}^{G}_{n}.$ Two $G$ -partial permutations $(h;(d_{1},\omega_{1}))$ and $(f;(d_{2},\omega_{2}))$ of $[n]$ are in the same conjugacy class if and only if there exists $(g;\sigma)\in G\wr\mathcal{S}_{n}$ such that $(g;\sigma)\cdot(h;(d_{1},\omega_{1}))=(f;(d_{2},\omega_{2})),$ that is $d_{2}=\sigma^{-1}(d_{1}),$ $\omega_{2}=\sigma\omega_{1}\sigma^{-1}$ and $f_{i}=g_{\omega_{1}^{-1}(\sigma(i))}h_{\sigma(i)}g^{-1}_{\sigma(i)}$ for any $i\in d_{2}.$

Example 4.1.

Let $\sigma=(2,3,6)(1,4)(5,7,9)(8)\in\mathcal{S}_{9},$ $d=\{2,4,5,6\}$ and $\omega=(2,5)(4,6).$ To obtain that

(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}=\big{(}(g_{6}h_{4}g_{4}^{-1},,g_{4}h_{6}g_{6}^{-1},,,g_{5}h_{2}g_{2}^{-1},,,g_{2}h_{5}g_{5}^{-1});(\{1,3,6,9\},(1,3)(6,9))\big{)},

we have to draw the diagram of $(g;\sigma)$ restricted to $\sigma^{-1}(d)$ then below it the diagram of $\big{(}h;(d,\omega)\big{)}$ then below it the diagram of $(g;\sigma)^{-1}$ restricted to $d$ as shown below

Then it would be easy to verify that $\operatorname{type}\Big{(}(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}\Big{)}=\operatorname{type}\Big{(}\big{(}h;(d,\omega)\big{)}\Big{)}$ since the cycle product $h_{5}h_{2}$ of the cycle $(2,5)$ of $\omega$ is conjugate to the cycle product $g_{2}h_{5}g_{5}^{-1}g_{5}h_{2}g_{2}^{-1}$ of the cycle $(6,9)$ of $\sigma\omega\sigma^{-1}$ and the cycle product $h_{6}h_{4}$ of the cycle $(4,6)$ of $\omega$ is conjugate to the cycle product $g_{4}h_{6}g_{6}^{-1}g_{6}h_{4}g_{4}^{-1}$ of the cycle $(1,3)$ of $\sigma\omega\sigma^{-1}.$

This shows that the conjugacy classes of $\mathfrak{P}^{G}_{n}$ can be indexed by the elements of the set $\mathcal{P}^{G_{\star}}_{\leq n}$ of families of partitions indexed by $G_{\star}$ with size less than or equal to $n.$ If $\Lambda=(\Lambda(c))_{c\in G_{\star}}\in\mathcal{P}^{G_{\star}}_{\leq n},$ the conjugacy class of $\mathfrak{P}^{G}_{n}$ associated to $\Lambda$ will be denoted $C_{\Lambda;n}$ and is defined by:

C_{\Lambda;n}:=\{x=(h;(d,\omega))\in\mathfrak{P}^{G}_{n}\text{ such that }|d|=|\Lambda|\text{ and }\operatorname{type}(x)=\Lambda\}.

Proposition 4.2.

If $\Lambda=(\Lambda(c))_{c\in G_{\star}}\in\mathcal{P}^{G_{\star}}_{\leq n},$ then:

|C_{\Lambda;n}|=\begin{pmatrix}n-|\Lambda|+m_{1}(\lambda(\{1_{G}\}))\\ m_{1}(\lambda(\{1_{G}\}))\end{pmatrix}|C_{\underline{\Lambda}_{n}}|.

Proof.

Consider the following mapping

\begin{array}[]{ccccc}\Theta&:&C_{\Lambda;n}&\to&C_{\underline{\Lambda}_{n}}\\ &&(g;(d,\omega))&\mapsto&(\widetilde{g};\widetilde{\omega}).\\ \end{array}

If $v,v^{{}^{\prime}}\in C_{\underline{\Lambda}_{n}}$ with $v\neq v^{{}^{\prime}},$ we have $\Theta^{-1}(v)\cap\Theta^{-1}(v^{{}^{\prime}})=\emptyset$ which implies that

(2)

|C_{\Lambda;n}|=\sum_{v\in C_{\underline{\Lambda}_{n}}}|\Theta^{-1}(v)|.

Let $(\sigma;p)\in C_{\underline{\Lambda}_{n}}$ and consider the set $\operatorname{supp}(\sigma;p)$ defined as follows:

\operatorname{supp}(\sigma;p)=\{i\in[n]\text{ such that }p(i)\neq i\text{ or }p(i)=i\text{ and }\sigma_{i}\neq 1_{G}\}.

Since $(\sigma;p)$ has type $\underline{\Lambda}_{n}$ then it would be clear that $|\operatorname{supp}(\sigma;p)|=|\Lambda|-m_{1}(\lambda(\{1_{G}\})).$ To make an element $(g;(d,\omega))\in\Theta^{-1}(\sigma;p),$ $(g;(d,\omega))$ must coincide with $(\sigma;p)$ on $\operatorname{supp}(\sigma;p)$ which necessarily implies that $\operatorname{supp}(\sigma;p)\subset d$ and we choose $m_{1}(\Lambda(\{1_{G}\}))=|d|-|\operatorname{supp}(\sigma;p)|$ fixed points for $(d,\omega)$ among the $n-|\Lambda|+m_{1}(\Lambda(\{1_{G}\}))$ fixed points of $\sigma.$ Thus for any $(\sigma;p)\in C_{\underline{\Lambda}_{n}}$ we have

|\Theta^{-1}(\sigma;p)|=\begin{pmatrix}n-|\Lambda|+m_{1}(\lambda(\{1_{G}\}))\\ m_{1}(\lambda(\{1_{G}\}))\end{pmatrix}.

Combining this formula with Equation (2) ends the proof. ∎

We extend the action of $G\wr\mathcal{S}_{n}$ on $\mathfrak{P}^{G}_{n}$ by linearity to an action on $\mathcal{B}^{G}_{n}:=\mathbb{C}[\mathfrak{P}^{G}_{n}],$ the algebra of the semigroup $\mathfrak{P}^{G}_{n},$ and we denote

\mathcal{A}^{G}_{n}:=\{b\in\mathcal{B}^{G}_{n}\text{ such that for any $(\sigma;p)\in G\wr\mathcal{S}_{n},$ }(\sigma;p)\cdot b=b\}

the sub-algebra of invariant elements under this action.

Proposition 4.3.

The surjective homomorphism

\begin{array}[]{ccccc}\psi&:&\mathfrak{P}^{G}_{n}&\to&G\wr\mathcal{S}_{n}\\ &&(h;(d,\omega))&\mapsto&(\widetilde{h};\widetilde{\omega})\\ \end{array}

is compatible with the action of $G\wr\mathcal{S}_{n}$ on $\mathfrak{P}^{G}_{n}.$

Proof.

We need to prove that for any $(g;\sigma)\in G\wr\mathcal{S}_{n}$ and any $\big{(}h;(d,\omega)\big{)}\in\mathfrak{P}^{G}_{n}$ we have:

\psi\big{(}(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}\big{)}=(g;\sigma)\cdot\psi\big{(}\big{(}h;(d,\omega)\big{)}\big{)}=(g;\sigma)(\widetilde{h};\widetilde{\omega})(g;\sigma)^{-1}.

We have $(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}=\big{(}f;(\sigma^{-1}(d),\sigma\omega\sigma^{-1})\big{)}$ where $f_{i}=g_{\omega^{-1}(\sigma(i))}h_{\sigma(i)}g^{-1}_{\sigma(i)}$ if $i\in\sigma^{-1}(d)$ which implies that $\psi\big{(}(g;\sigma)\cdot\big{(}h;(d,\omega)\big{)}\big{)}=\big{(}\widetilde{f};\widetilde{\sigma\omega\sigma^{-1}}\big{)}$ with

\widetilde{\sigma\omega\sigma^{-1}}(a)=\left\{\begin{array}[]{ll}(\sigma\omega\sigma^{-1})(a)&\qquad\mathrm{if}\quad a\in\sigma^{-1}(d)\\ a&\qquad\mathrm{otherwise}\\ \end{array}\right.\text{ and }\widetilde{f}_{i}=\left\{\begin{array}[]{ll}f_{i}&\qquad\mathrm{if}\quad i\in\sigma^{-1}(d)\\ 1_{G}&\qquad\mathrm{otherwise}.\\ \end{array}\right.

On the other hand $(g;\sigma)(\widetilde{h};\widetilde{\omega})(g;\sigma)^{-1}=(r;\sigma\widetilde{\omega}\sigma^{-1})$ with $r_{i}=g_{\widetilde{\omega}^{-1}(\sigma(i))}\widetilde{h}_{\sigma(i)}g^{-1}_{\sigma(i)}.$ If $i\notin\sigma^{-1}(d)$ then $\sigma(i)\notin d$ which implies that $\widetilde{\omega}^{-1}(\sigma(i))=\sigma(i)$ and $\widetilde{h}_{\sigma(i)}=1_{G}$ which results in $r_{i}=1_{G}.$ If $i\in\sigma^{-1}(d)$ then $r_{i}=f_{i}.$ Thus $\widetilde{f}=r$ and it is easy to check that $\widetilde{\sigma\omega\sigma^{-1}}=\sigma\widetilde{\omega}\sigma^{-1}$ which ends the proof. ∎

The surjective homomorphism $\psi$ can be extended to a surjective homomorphism of algebras $\psi:\mathcal{B}^{G}_{n}\rightarrow\mathbb{C}[G\wr\mathcal{S}_{n}]$ and Proposition 4.3 implies that

\psi(\mathcal{A}^{G}_{n})=Z(\mathbb{C}[\mathcal{S}_{n}]).

If $\Lambda=(\Lambda(c))_{c\in G_{\star}}\in\mathcal{P}^{G_{\star}}_{\leq n},$ let us denote by ${\bf{C}}_{\Lambda;n}$ the following formal sum:

{\bf C}_{\Lambda;n}=\sum_{(h;(d,\omega))\in C_{\Lambda;n}}(h;(d,\omega)).

The elements of the family $({\bf{C}}_{\Lambda;n})_{\Lambda\in\mathcal{P}^{G_{\star}}_{\leq n}}$ form a basis for the algebra $\mathcal{A}^{G}_{n}$ and if $\Lambda\in\mathcal{P}^{G_{\star}}_{\leq n},$ then by Proposition 4.2 we have:

(3)

\psi({\bf{C}}_{\Lambda;n})=\begin{pmatrix}n-|\Lambda|+m_{1}(\lambda(\{1_{G}\}))\\ m_{1}(\lambda(\{1_{G}\}))\end{pmatrix}{\bf C}_{\underline{\Lambda}_{n}}.

5. Action of $G\wr\mathcal{S}_{\infty}$ on $\mathcal{B}^{G}_{\infty}$

Denote by $\mathcal{B}^{G}_{\infty}$ the projective limit of the algebras $\mathcal{B}^{G}_{n}$ with respect to the homomorphism $\varphi_{n}:\mathcal{B}^{G}_{n+1}\rightarrow\mathcal{B}^{G}_{n}$ defined by:

\varphi_{n}(h;(d,\omega))=\left\{\begin{array}[]{ll}(h;(d,\omega))&\qquad\mathrm{if}\quad d\subset[n],\\ 0&\qquad\mathrm{otherwise.}\\ \end{array}\right.

If $d$ is a finite subset of $\mathbb{N},$ we define $G^{\infty}_{d}$ to be the set of vectors $x$ with infinite number of coordinates such that $x_{i}\in G$ whenever $i\in d$ and $x_{i}$ is left blank otherwise. An element $b\in\mathcal{B}^{G}_{\infty}$ can be canonically written:

b=\sum_{k=0}^{\infty}\sum_{|d|=k}\sum_{\omega\in\mathcal{S}_{d}}\sum_{h\in G^{\infty}_{d}}b_{(h;(d,\omega))}(h;(d,\omega)),

where the $b_{(h;(d,\omega))}$ ’s are complex numbers. Consider the group $\mathcal{S}_{\infty}$ of permutations of $\mathbb{N}$ with finite support. Let the group $G\wr\mathcal{S}_{\infty}$ acts on $\mathcal{B}^{G}_{\infty}$ by conjugation and denote by $\mathcal{A}^{G}_{\infty}$ the sub-algebra of invariant elements of $\mathcal{B}^{G}_{\infty}$ under this action. An element $b\in\mathcal{B}^{G}_{\infty}$ is in $\mathcal{A}^{G}_{\infty}$ if and only if:

b_{(h;(d,\omega))}=b_{(f;(\sigma^{-1}(d),\sigma\omega\sigma^{-1}))}\text{ for any }(g;\sigma)\in G\wr\mathcal{S}_{\infty},

where $f_{i}=g_{\omega_{1}^{-1}(\sigma(i))}h_{\sigma(i)}g^{-1}_{\sigma(i)}$ if $i\in\sigma^{-1}(d)$ and $f_{i}$ is left blank if $i\notin\sigma^{-1}(d).$

Definition 5.1.

A $G$ -partial permutation of $\mathbb{N}$ is a pair $(h;(d,\omega))$ where $d\subsetneq\mathbb{N}$ is a finite subset of $\mathbb{N},$ $\omega\in\mathcal{S}_{d}$ and $h\in G_{d}^{\infty}.$

We denote by $\mathfrak{P}_{\infty}^{G}$ the set of all $G$ -partial permutations of $\mathbb{N}.$ For a family of partitions $\Lambda=(\lambda(c))_{c\in G_{\star}}\in\mathcal{P}^{G_{\star}}$ indexed by $G_{\star},$ we define $C_{\Lambda;\infty}$ as follows:

C_{\Lambda;\infty}=\{(h;(d,\omega))\in\mathfrak{P}_{\infty}^{G}\text{ such that }|d|=|\Lambda|\text{ and }\operatorname{type}(h;(d,\omega))=\Lambda\}.

Any element in $\mathcal{A}^{G}_{\infty}$ can be written in a unique way as an infinite linear combination of elements $({\bf C}_{\Lambda;\infty})_{\Lambda\in\mathcal{P}^{G}},$ where

{\bf C}_{\Lambda;\infty}=\sum_{(h;(d,\omega))\in C_{\Lambda;\infty}}(h;(d,\omega)).

Let $\Lambda$ and $\Delta$ be two families of partitions in $\mathcal{P}^{G_{\star}},$ the structure coefficients $k_{\Lambda\Delta}^{\Gamma}$ of the algebra $\mathcal{A}^{G}_{\infty}$ are defined by:

(4)

{\bf C}_{\Lambda;\infty}{\bf C}_{\Delta;\infty}=\sum_{\Gamma\in\mathcal{P}^{G_{\star}}}k_{\Lambda\Delta}^{\Gamma}{\bf C}_{\Gamma;\infty}.

Proposition 5.2.

The function $\operatorname{Fil}$ defined by $\operatorname{Fil}({\bf C}_{\Lambda;\infty})=|\Lambda|,$ for any $\Lambda\in\mathcal{P}^{G_{\star}},$ is a filtration on $\mathcal{A}^{G}_{\infty}.$

Proof.

We need to prove that $\operatorname{Fil}({\bf C}_{\Lambda;\infty}{\bf C}_{\Delta;\infty})\leq\operatorname{Fil}({\bf C}_{\Lambda;\infty})+\operatorname{Fil}({\bf C}_{\Delta;\infty})$ for any two families of partitions $\Lambda,\Delta\in\mathcal{P}^{G_{\star}}.$ For this let $\Gamma$ be a family of partitions in $\mathcal{P}^{G_{\star}}$ for which $k_{\Lambda\Delta}^{\Gamma}$ of Equation (4) is a non-zero coefficient. By its definition, $k_{\Lambda\Delta}^{\Gamma}$ counts the number of pairs of $G$ -partial permutations $\big{(}(g_{1};(d_{1},\omega_{1})),(g_{2};(d_{2},\omega_{2}))\big{)}\in C_{\Lambda;\infty}\times C_{\Delta;\infty}$ such that

(g_{1};(d_{1},\omega_{1})).(g_{2};(d_{2},\omega_{2}))=(g;(d,\omega))

where $(g;(d,\omega))$ is a fixed $G$ -partial permutation belonging to $C_{\Gamma;\infty}.$ It would be then sufficient to remark that, when multiplying $(g_{1};(d_{1},\omega_{1}))$ by $(g_{2};(d_{2},\omega_{2})),$ the permutation $\widetilde{\omega}_{1_{|d_{1}\cup d_{2}}}\widetilde{\omega}_{2_{|d_{1}\cup d_{2}}}$ acts on at most $|d_{1}\cup d_{2}|$ elements. This means that each family of partitions $\Gamma$ that appears in the sum of Equation (4) must satisfy

(5)

\max(|\Lambda|,|\Delta|)\leq|\Gamma|\leq|\Lambda|+|\Delta|,

which ends the proof. ∎

Remark 5.3.

More filtrations on $\mathcal{A}^{G}_{\infty}$ may exist as suggested by [3]. In this paper, we will only use the above proved one.

We denote by $\operatorname{Proj}_{n}$ the natural projection homomorphism between $\mathcal{B}^{G}_{\infty}$ and $\mathcal{B}^{G}_{n}$ defined on the generating element of $\mathcal{B}^{G}_{\infty}$ by

\operatorname{Proj}_{n}(h;(d,\omega))=\left\{\begin{array}[]{ll}(h;(d,\omega))&\qquad\mathrm{if}\quad d\subset[n],\\ 0&\qquad\mathrm{otherwise.}\\ \end{array}\right.

If $\Lambda\in\mathcal{P}^{G_{\star}},$ then we have:

\operatorname{Proj}_{n}({\bf C}_{\Lambda;\infty})=\left\{\begin{array}[]{ll}{\bf C}_{\Lambda;n}&\qquad\mathrm{if}\quad|\Lambda|\leq n,\\ 0&\qquad\mathrm{otherwise.}\\ \end{array}\right.

By Equation (4), $k_{\Lambda\Delta}^{\Gamma}$ are also the structure coefficients of the algebra $\mathcal{A}^{G}_{n}:$

{\bf C}_{\Lambda;n}{\bf C}_{\Delta;n}=\sum_{\Gamma\in\mathcal{P}^{G_{\star}}_{\leq n},\atop{|\Gamma|\leq|\Lambda|+|\Delta|}}k_{\Lambda\Delta}^{\Gamma}{\bf C}_{\Gamma;n},

where $\Lambda$ and $\Delta$ are two families of partitions belonging to $\mathcal{P}^{G_{\star}}_{\leq n}.$ In other words, the structure coefficients $k_{\Lambda\Delta}^{\Gamma}$ of $\mathcal{A}^{G}_{\infty}$ do not depend on $n$ and they are the structure coefficients of $\mathcal{A}^{G}_{n}$ for any $n.$

6. Polynomiality of the structure coefficients of $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$

In this section we will prove our main result in Theorem 6.2. For this we will need the following lemma.

Lemma 6.1.

If $\Gamma\in\mathcal{P}^{G_{\star}}_{\leq n}$ then we have ${\bf C}_{\underline{\Gamma}_{n}}={\bf C}_{\underline{\Gamma^{j}}_{n}}$ for any $0\leq j\leq n-|\Gamma|,$ where $\Gamma^{j}=(\gamma^{j}(c))_{c\in G_{\star}}$ is the family of partitions of size $|\Gamma|+j$ indexed by $G_{\star}$ and defined by $\gamma^{j}(\{1_{G}\})=\gamma(\{1_{G}\})\cup(1^{j})$ and $\gamma^{j}(c)=\gamma(c)$ if $c\neq\{1_{G}\}.$

Theorem 6.2.

Let $\Lambda,\Delta$ and $\Gamma$ be three proper families of partitions indexed by $G_{\star}$ and let $n$ be a natural number with $n\geq|\Lambda|,|\Delta|,|\Gamma|.$ The structure coefficient $c_{\Lambda\Delta}^{\Gamma}(n)$ is a polynomial in $n$ with degree $\displaystyle\max_{\overset{0\leq j\leq n-|\Gamma|}{k_{\Lambda\Delta}^{\Gamma^{j}}\neq 0}}j$ that can be written:

c_{\Lambda\Delta}^{\Gamma}(n)=\sum_{j=0}^{n-|\Gamma|}k_{\Lambda\Delta}^{\Gamma^{j}}\begin{pmatrix}n-|\Gamma|\\ j\end{pmatrix},

where the coefficients $k_{\Lambda\Delta}^{\Gamma^{j}}$ are independant integers of $n.$

Proof.

If $\Lambda$ and $\Delta$ are two proper partitions then by Equation (3) we have $\psi\big{(}{\bf C}_{\Lambda;n}\big{)}={\bf C}_{\underline{\Lambda}_{n}}$ and $\psi\big{(}{\bf C}_{\Delta;n}\big{)}={\bf C}_{\underline{\Delta}_{n}}.$ Recall the following equation in $\mathcal{A}^{G}_{n}:$

{\bf C}_{\Lambda;n}{\bf C}_{\Delta;n}=\sum_{\Gamma\in\mathcal{P}^{G_{\star}}_{\leq n},\atop{|\Gamma|\leq|\Lambda|+|\Delta|}}k_{\Lambda\Delta}^{\Gamma}{\bf C}_{\Gamma;n},

Apply $\psi$ to get:

{\bf C}_{\underline{\Lambda}_{n}}{\bf C}_{\underline{\Delta}_{n}}=\sum_{\Gamma\in\mathcal{P}^{G_{\star}}_{\leq n},\atop{|\Gamma|\leq|\Lambda|+|\Delta|}}k_{\Lambda\Delta}^{\Gamma}\begin{pmatrix}n-|\Gamma|+m_{1}(\gamma(\{1_{G}\}))\\ m_{1}(\gamma(\{1_{G}\}))\end{pmatrix}{\bf C}_{\underline{\Gamma}_{n}}.

Thus, by Lemma 6.1, the right hand side summation of the above equation can be written:

\sum_{\Gamma\in\mathcal{PP}^{G_{\star}}_{\leq n},\atop{|\Gamma|\leq|\Lambda|+|\Delta|}}\left[\sum_{j=0}^{n-|\Gamma|}k_{\Lambda\Delta}^{\Gamma^{j}}\begin{pmatrix}n-|\Gamma^{j}|+m_{1}(\gamma^{j}(\{1_{G}\}))\\ m_{1}(\gamma^{j}(\{1_{G}\}))\end{pmatrix}\right]{\bf C}_{\underline{\Gamma}_{n}}.

After simplification, we obtain:

\sum_{\Gamma\in\mathcal{PP}^{G_{\star}}_{\leq n},\atop{|\Gamma|\leq|\Lambda|+|\Delta|}}\left[\sum_{j=0}^{n-|\Gamma|}k_{\Lambda\Delta}^{\Gamma^{j}}\begin{pmatrix}n-|\Gamma|\\ j\end{pmatrix}\right]{\bf C}_{\underline{\Gamma}_{n}},

which ends the proof. ∎

Example 6.3.

Let $p>2$ be a prime number and consider $G=\mathbb{Z}_{p}.$ Recall that the type of $x=(g;p)\in\mathbb{Z}_{p}\wr\mathcal{S}_{n}$ is a $p$ -vector of partitions $\Lambda=(\lambda_{0},\lambda_{1},\ldots,\lambda_{p-1})$ where each partition $\lambda_{i}$ is formed out of cycles $c$ of $p$ whose cycle product equals $i.$ If $0\leq i\leq p-1$ and $\rho$ is a partition then we denote by $\rho^{i}$ the $p$ -vector of partitions $(\lambda_{0},\lambda_{1},\ldots,\lambda_{p-1})$ where $\lambda_{i}=\rho$ and $\lambda_{s}=\emptyset$ if $s\neq i.$ If $\rho$ is a proper partition then it would be clear that $\rho^{i}$ is a proper family of partitions. We have:

{\bf C}_{(1)^{i};\infty}=\sum_{j\in\mathbb{N}^{\star}}\big{(}~{}\widehat{i}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)},

where $\widehat{i}^{j}\in G^{\infty}_{\{j\}}$ is the vector $x$ with infinite number of coordinates such that $x_{j}=i$ and $x_{v}$ is left blank if $v\neq j.$ It would be clear that $(1)^{i}$ is proper if and only if $0<i\leq p-1.$ Suppose that $j<r$ then we have

\big{(}~{}\widehat{i}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)}\big{(}~{}\widehat{t}^{r};(\{r\},\operatorname{Id}_{\{r\}})~{}\big{)}=\big{(}~{}\widehat{i}^{j}+\widehat{t}^{r};(\{j,r\},\operatorname{Id}_{\{j,r\}})~{}\big{)}.

This can be deduced by drawing the multiplication diagram as below:

In addition, we have:

\big{(}~{}\widehat{i}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)}\big{(}~{}\widehat{t}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)}=\big{(}~{}\widehat{i+t}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)},

where the sum $i+t$ is taken modulo $p.$ Thus, if $0<i\leq p-1,$ then

(6)

{\bf C}_{(1)^{i};\infty}{\bf C}_{(1)^{t};\infty}=\left\{\begin{array}[]{ll}{\bf C}_{(1)^{2i};\infty}+2{\bf C}_{(1^{2})^{i};\infty}&\qquad\mathrm{if}\quad t=i,\\ {\bf C}_{(1)^{i+t};\infty}+2{\bf C}_{(1)^{i}\cup(1)^{t};\infty}&\qquad\mathrm{otherwise,}\\ \end{array}\right.

where $(1)^{i}\cup(1)^{t}$ is the $p$ -vector of partitions $(\lambda_{0},\lambda_{1},\ldots,\lambda_{p-1})$ with $\lambda_{i}=(1),$ $\lambda_{t}=(1)$ and $\lambda_{s}=\emptyset$ if $s\notin\{i,t\}.$ The coefficient $2$ of ${\bf C}_{(1)^{i}\cup(1)^{t};\infty}$ in the right hand side of Equation (6) is due to the fact that both products $\big{(}~{}\widehat{i}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)}\big{(}~{}\widehat{t}^{r};(\{r\},\operatorname{Id}_{\{r\}})~{}\big{)}$ and $\big{(}~{}\widehat{t}^{r};(\{r\},\operatorname{Id}_{\{r\}})~{}\big{)}\big{(}~{}\widehat{i}^{j};(\{j\},\operatorname{Id}_{\{j\}})~{}\big{)}$ yields the same $\mathbb{Z}_{p}$ -partial permutation $\big{(}~{}\widehat{i}^{j}+\widehat{t}^{r};(\{j,r\},\operatorname{Id}_{\{j,r\}})~{}\big{)}.$ If $n\geq 2,$ by applying $\theta_{n}$ then $\psi$ on Equation (6) we obtain:

{\bf C}_{\underline{(1)^{i}}_{n}}{\bf C}_{\underline{(1)^{i}}_{n}}={\bf C}_{\underline{(1)^{2i}}_{n}}+2{\bf C}_{\underline{(1^{2})^{i}}_{n}}\text{if $i\neq 0$},

{\bf C}_{\underline{(1)^{i}}_{n}}{\bf C}_{\underline{(1)^{t}}_{n}}={\bf C}_{\underline{(1)^{i+t}}_{n}}+2{\bf C}_{\underline{(1)^{i}\cup(1)^{t}}_{n}}\text{if $i,t\neq 0,$ $i\neq t$ and $i+t\neq 0$ (mod $p$)},

and

{\bf C}_{\underline{(1)^{i}}_{n}}{\bf C}_{\underline{(1)^{t}}_{n}}={n\choose 1}{\bf C}_{\underline{(1)^{i+t}}_{n}}+2{\bf C}_{\underline{(1)^{i}\cup(1)^{t}}_{n}}\text{ if $i,t\neq 0,$ $i\neq t$ and $i+t=0$ (mod $p$)}.

7. Irreducible characters of $G\wr\mathcal{S}_{n}$ and symmetric functions

In this section we will recall all the necessary definitions and results from the theory of summetric functions in order to prove that the algebra $\mathcal{A}_{\infty}^{G}$ is isomorphic to an algebra of shifted symmetric functions on $|G^{\star}|$ alphabets.

7.1. The case $G=\{1_{G}\}$

When $G=\{1_{G}\},$ the wreath product $G\wr\mathcal{S}_{n}$ is isomorphic to the symmetric group $\mathcal{S}_{n}.$ The irreducible $\mathcal{S}_{n}$ -modules are indexed by partitions of $n.$ If $\lambda\in\mathcal{P}_{n},$ we will denote by $V^{\lambda}$ its associated irreducible $\mathcal{S}_{n}$ -module and by $\chi^{\lambda}$ its character. It is well known that both the power sum functions $(p_{\lambda})_{\lambda}$ and the Schur functions $(s_{\lambda})_{\lambda},$ indexed by partitions, are basis families for the algebra $\mathrm{A}$ of symmetric functions. The transition matrix between these two bases is given by the following formula of Frobenius:

(7)

p_{\delta}=\sum_{\rho\atop{|\rho|=|\delta|}}\chi^{\rho}_{\delta}s_{\rho},

where $\chi^{\rho}_{\delta}$ denotes the value of the character $\chi^{\rho}$ on any permutation of cycle-type $\delta.$

A shifted symmetric function $f$ in infinitely many variables $(x_{1},x_{2},\ldots)$ is a family $(f_{i})_{i>1}$ that satisfies the following two properties:

1.

$f_{i}$ is a symmetric polynomial in $(x_{1}-1,x_{2}-2,\ldots,x_{i}-i).$
2.

$f_{i+1}(x_{1},x_{2},\ldots,x_{i},0)=f_{i}(x_{1},x_{2},\ldots,x_{i}).$

The set of all shifted symmetric functions is an algebra which we shall denote $\mathrm{A}^{\#}.$ In [7], Okounkov and Olshanski gave a linear isomorphism $\varphi:\mathrm{A}\rightarrow\mathrm{A}^{\#}.$ For any partition $\lambda,$ the images of the power sum function $p_{\lambda}$ and the Schur function $s_{\lambda}$ by $\varphi$ are the shifted power symmetric function $p^{\#}_{\lambda}$ and the shifted Schur function $s^{\#}_{\lambda}.$ By applying $\varphi$ to the Frobenius relation given in Equation (7), we get:

(8)

p^{\#}_{\delta}=\sum_{\rho\atop{|\rho|=|\delta|}}\chi^{\rho}_{\delta}s^{\#}_{\rho}.

If $f\in\mathrm{A}^{\#}$ and if $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{l})$ is a partition, we denote by $f(\lambda)$ the value $f_{l}(\lambda_{1},\lambda_{2},\cdots,\lambda_{l}).$ By [7], any shifted symmetric function is determined by its values on partitions. The vanishing characterization of the shifted symmetric functions given in [7] states that $s_{\rho}^{\#}$ is the unique shifted symmetric function of degree at most $|\rho|$ such that

(9)

s^{\#}_{\rho}(\lambda)=\left\{\begin{array}[]{ll}\frac{(|\lambda|\downharpoonright|\rho|)}{\dim\lambda}f^{\lambda/\rho}&\text{ if }\rho\subseteq\lambda\\ 0&\text{ otherwise }\\ \end{array}\right.

where $(|\lambda|\downharpoonright|\rho|):=|\lambda|(|\lambda|-1)\cdots(|\lambda|-|\rho|+1)$ is the falling factorial and $f^{\lambda/\rho}$ is the number of skew standard tableaux of shape $\lambda/\rho.$ Using the following branching rule for characters of the symmetric groups

(10)

\chi^{\lambda}_{\rho\cup(1^{|\lambda|-|\rho|})}=\sum_{\nu;|\nu|=|\rho|}f^{\lambda/\nu}\chi^{\nu}_{\rho},

one can verify using formulas (8) and (9) that

p^{\#}_{\delta}(\lambda)=\left\{\begin{array}[]{ll}\frac{(|\lambda|\downharpoonright|\delta|)}{\dim\lambda}\chi^{\lambda}_{\underline{\delta}_{|\lambda|}}&\text{ if }|\lambda|\geq|\delta|\\ 0&\text{ otherwise }.\\ \end{array}\right.

Using this formula, Ivanov and Kerov showed in [3, Theorem 9.1] that the mapping $F:\mathcal{A}_{\infty}^{\{1_{G}\}}\rightarrow\mathrm{A}^{\#}$ defined on the basis elements of $\mathcal{A}_{\infty}^{\{1_{G}\}}$ by

F(\mathbf{C}_{\delta})=z_{\delta}^{-1}p^{\#}_{\delta}

is an isomorphism of algebras.

7.2. The general case

We refer to [5, Appendix B] for the results of the representation theory of wreath products presented in this section. Let $(P_{r}(c))_{r\geq 1,c\in G_{\star}}$ be a family of independent indeterminates over $\mathbb{C}.$ For each $c\in G_{\star},$ we may think of $P_{r}(c)$ as the $r^{th}$ power sum in a sequence of variables $x_{c}=(x_{ic})_{i\geq 1}.$ Let us denote by $\mathrm{A}^{G}$ the algebra over $\mathbb{C}$ with algebraic basis the elements $P_{r}(c)$

\mathrm{A}^{G}:=\mathbb{C}[P_{r}(c);r\geq 1,c\in G_{\star}].

If $\rho=(\rho_{1},\rho_{2},\cdots,\rho_{l})$ is an arbitrary partition and $c\in G_{\star},$ we define $P_{\rho}(c)$ to be the product of $P_{\rho_{i}}(c),$

P_{\rho}(c):=P_{\rho_{1}}(c)P_{\rho_{2}}(c)\cdots P_{\rho_{l}}(c).

The family $(P_{\Lambda})_{\Lambda\in\mathcal{P}^{G_{\star}}},$ where

P_{\Lambda}:=\prod_{c\in G_{\star}}P_{\Lambda(c)}(c),

forms a linear basis for $\mathrm{A}^{G}.$ That is any element $f\in\mathrm{A}^{G}$ can be written $f=\sum\limits_{\Lambda\in\mathcal{P}^{G_{\star}}}f_{\Lambda}P_{\Lambda}$ where all but a finite number of the coefficients $f_{\Lambda}\in\mathbb{C}$ are zero. If we assign degree $r$ to $P_{r}(c),$ then

\mathrm{A}^{G}=\bigoplus_{n\geq 0}\mathrm{A}_{n}^{G}

is a graded $\mathbb{C}$ -algebra where $\mathrm{A}_{n}^{G}$ is the algebra spanned by all $P_{\Lambda}$ where $\Lambda\in\mathcal{P}^{G_{\star}}_{n}.$ The algebra $\mathrm{A}^{G}$ can be equipped with a hermitian scalar product defined by

<f,g>=\sum_{\Lambda}f_{\Lambda}\bar{g}_{\Lambda}Z_{\Lambda}

for any two elements $f=\sum\limits_{\Lambda\in\mathcal{P}^{G_{\star}}}f_{\Lambda}P_{\Lambda}$ and $g=\sum\limits_{\Lambda\in\mathcal{P}^{G_{\star}}}g_{\Lambda}P_{\Lambda}$ of $\mathcal{A}^{G}.$ In particular, we have:

<P_{\Lambda},P_{\Gamma}>=\delta_{\Lambda,\Gamma}Z_{\Lambda},

where $\delta_{\Lambda,\Gamma}$ is the Kronecker symbol.

For each irreducible character $\gamma\in G^{\star}$ and each $r\geq 1,$ let

P_{r}(\gamma):=\sum_{c\in G_{\star}}\xi_{c}^{-1}\gamma(c)P_{r}(c),

where $\gamma(c)$ is the value of the character $\gamma$ on an element of the conjugacy class $c.$ By the orthogonality of the characters of $G,$

<\gamma,\rho>:=\frac{1}{|G|}\sum_{g\in G}\gamma(g)\rho(g)=\sum_{c\in G_{\star}}\xi_{c}^{-1}\gamma(c)\rho(c)=\delta_{\gamma,\delta},

we can write

P_{r}(c)=\sum_{\gamma\in G^{\star}}\overline{\gamma(c)}P_{r}(\gamma).

We may think of $P_{r}(\gamma)$ as the $r^{th}$ power sum in a new sequence of variables $y_{\gamma}=(y_{i\gamma})_{i\geq 1}$ and denote by $s_{\rho}(\gamma)$ the schur function $s_{\rho}$ associated to the partition $\rho$ on the sequence of variables $(y_{i\gamma})_{i\geq 1}.$ Now, for any family of partitions $\Lambda\in\mathcal{P}^{G^{\star}},$ define

S_{\Lambda}:=\prod_{\gamma\in G^{\star}}s_{\Lambda(\gamma)}(\gamma).

The family $(S_{\Lambda})_{\Lambda\in\mathcal{P}^{G^{\star}}}$ is an orthonormal basis of $\mathrm{A}^{G},$ see [5, (7.4)].

Let $V^{\gamma}$ be the irreducible $G$ -module associated to $\gamma\in G^{\star}.$ The group $G\wr\mathcal{S}_{n}$ acts on the $n^{th}$ tensor power $T^{n}(V^{\gamma})=V^{\gamma}\otimes V^{\gamma}\cdots\otimes V^{\gamma}$ as follows:

(g;p)\cdot(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{n}):=g_{1}v_{p^{-1}(1)}\otimes\cdots\otimes g_{n}v_{p^{-1}(n)},

where $(g;p)\in G\wr\mathcal{S}_{n}$ and $v_{1},v_{2},\cdots,v_{n}\in V^{\gamma}.$ Let us denote by $\eta_{n}(\gamma)$ the character of this representation of $G\wr\mathcal{S}_{n}.$ From [5, (8.2)], if $x\in G\wr\mathcal{S}_{n}$ has type $\Lambda\in\mathcal{P}^{G_{\star}}_{n},$ then:

\eta_{n}(\gamma)(x)=\prod_{c\in G_{\star}}{\gamma(c)}^{l(\Lambda(c))}.

For any partition $\mu$ of $m$ and each $\gamma\in G^{\star},$ we define

X^{\mu}(\gamma):=\det(\eta_{\mu_{i}-i+j}({\gamma})).

We can extend this definition to families of partitions $\Lambda\in\mathcal{P}^{G^{\star}}_{n}.$ If $\Lambda\in\mathcal{P}^{G^{\star}}_{n},$ let

X^{\Lambda}:=\prod_{\gamma\in G^{\star}}X^{\Lambda(\gamma)}(\gamma).

The family $(X^{\Lambda})_{\Lambda\in\mathcal{P}^{G^{\star}}_{n}}$ is a full list of irreducible characters of $G\wr\mathcal{S}_{n}.$ For any two families of partitions $\Lambda\in\mathcal{P}^{G^{\star}}_{n}$ and $\Delta\in\mathcal{P}^{G_{\star}}_{n},$ let us denote by $X^{\Lambda}_{\Delta}$ the value of the character $X^{\Lambda}$ on any of the elements of the conjugacy class $C_{\Delta}.$ By [5, page 177], we have the following three important identities

X^{\Lambda}_{\Delta}=<S_{\Lambda},P_{\Delta}>,

S_{\Lambda}=\sum_{\Gamma\in\mathcal{P}^{G_{\star}}_{n}}Z_{\Gamma}^{-1}X^{\Lambda}_{\Gamma}P_{\Gamma}

and

P_{\Delta}=\sum_{\Sigma\in\mathcal{P}^{G^{\star}}_{n}}\overline{X^{\Sigma}_{\Lambda}}S_{\Sigma}.

Let us consider the algebra $\mathrm{A}^{G\#}$ isomorphic to $\mathrm{A}^{G}$ defined using the shifted symmetric functions. It has a basis formed by the shifted functions $P^{\#}_{\Delta}$ defined by

P^{\#}_{\Delta}:=\prod_{c\in G_{\star}}P^{\#}_{\Delta(c)}(c),

for any $\Delta\in\mathcal{P}^{G_{\star}}.$ For any family of partitions $\Lambda\in\mathcal{P}^{G_{\star}},$ we set

P^{\#}_{\Delta}(\Lambda):=\prod_{c\in G_{\star}}P^{\#}_{\Delta(c)}(c)(\Lambda(c)).

Theorem 7.1.

The linear map $F^{G}:\mathcal{A}_{\infty}^{G}\longrightarrow\mathrm{A}^{G\#}$ defined by

F^{G}(\mathbf{C}_{\Delta;\infty})=\frac{|G|^{|\Delta|}}{Z_{\Delta}}P^{\#}_{\Delta}

is an isomorphism of algebras.

Proof.

Let $\Lambda\in\mathcal{P}^{G^{\star}}_{n}$ and consider the composition $F_{\Lambda}^{G}:=\frac{X^{\Lambda}}{\dim\Lambda}\circ\psi\circ\operatorname{Proj}_{|\Lambda|}$ of morphisms. Let us see how $F_{\Lambda}^{G}$ acts on the basis elements of $\mathcal{A}_{\infty}^{G}.$ If $\Delta\in\mathcal{P}^{G_{\star}}$ with $|\Delta|>|\Lambda|,$ it would be clear then that $F_{\Lambda}^{G}({\bf C}_{\Delta;\infty})=0.$ Suppose now that $|\Lambda|\geq|\Delta|,$ we have the following equalities:

$\displaystyle\left(\frac{X^{\Lambda}}{\dim\Lambda}\circ\psi\circ\operatorname{Proj}_{\|\Lambda\|}\right)(\mathbf{C}_{\Delta;\infty})$	$\displaystyle=$	$\displaystyle\frac{X^{\Lambda}}{\dim\Lambda}\left({\|\Lambda\|-\|\Delta\|+m_{1}(\Delta(\{1_{G}\}))\choose m_{1}(\Delta(\{1_{G}\}))}\mathbf{C}_{\underline{\Delta}_{\|\Lambda\|}}\right)$
	$\displaystyle=$	$\displaystyle{\|\Lambda\|-\|\Delta\|+m_{1}(\Delta(\{1_{G}\}))\choose m_{1}(\Delta(\{1_{G}\}))}\frac{n!\|G\|^{n}}{Z_{\underline{\Delta}_{\|\Lambda\|}}\dim\Lambda}X^{\Lambda}_{\underline{\Delta}_{\|\Lambda\|}}$
	$\displaystyle=$	$\displaystyle\frac{(\|G\|)^{\|\Delta\|}}{Z_{\Delta}}\frac{(\|\Lambda\|\downharpoonright\|\Delta\|)}{\dim\Lambda}X^{\Lambda}_{\underline{\Delta}_{\|\Lambda\|}}$
	$\displaystyle=$	$\displaystyle\frac{(\|G\|)^{\|\Delta\|}}{Z_{\Delta}}P^{\#}_{\Delta}(\Lambda)$
	$\displaystyle=$	$\displaystyle F^{G}(\mathbf{C}_{\Delta;\infty})(\Lambda)$

∎

References

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The algebra of conjugacy classes of the wreath product of a finite group with the symmetric group

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

2. Algebra of the conjugacy classes of G≀𝒮nG\wr\mathcal{S}_{n}

2.1. Partitions

2.2. Conjugacy classes of G≀𝒮nG\wr\mathcal{S}_{n}

Example 2.1.

2.3. The center of the group G≀𝒮nG\wr\mathcal{S}_{n} algebra

3. GG-partial permutations

Definition 3.1.

Example 3.2.

Example 3.3.

4. Action of G≀𝒮nG\wr\mathcal{S}_{n} on 𝔓nG\mathfrak{P}^{G}_{n}

Example 4.1.

Proposition 4.2.

Proof.

Proposition 4.3.

Proof.

5. Action of G≀𝒮∞G\wr\mathcal{S}_{\infty} on ℬ∞G\mathcal{B}^{G}_{\infty}

Definition 5.1.

Proposition 5.2.

Proof.

Remark 5.3.

6. Polynomiality of the structure coefficients of Z(ℂ[G≀𝒮n])Z(\mathbb{C}[G\wr\mathcal{S}_{n}])

Lemma 6.1.

Theorem 6.2.

Proof.

Example 6.3.

7. Irreducible characters of G≀𝒮nG\wr\mathcal{S}_{n} and symmetric functions

7.1. The case G={1G}G=\{1_{G}\}

7.2. The general case

Theorem 7.1.

Proof.

References

The algebra of conjugacy classes of the wreath product
of a finite group with the symmetric group

2. Algebra of the conjugacy classes of $G\wr\mathcal{S}_{n}$

2.2. Conjugacy classes of $G\wr\mathcal{S}_{n}$

2.3. The center of the group $G\wr\mathcal{S}_{n}$ algebra

3. $G$ -partial permutations

4. Action of $G\wr\mathcal{S}_{n}$ on $\mathfrak{P}^{G}_{n}$

5. Action of $G\wr\mathcal{S}_{\infty}$ on $\mathcal{B}^{G}_{\infty}$

6. Polynomiality of the structure coefficients of $Z(\mathbb{C}[G\wr\mathcal{S}_{n}])$

7. Irreducible characters of $G\wr\mathcal{S}_{n}$ and symmetric functions

7.1. The case $G=\{1_{G}\}$