Stability of the centers of group algebras of general affine groups $GA_{n}(q)$

In [FH] by Farahat and Higman, a fundamental stability result for the centers of the integral group algebras $\mathbb{Z}[S_{n}]$ of the symmetric groups $S_{n}$ was established. By introducing a conjugation-invariant reflection length for permutations, the center of $\mathbb{Z}[S_{n}]$ admits a filtered algebra structure. It is proved in [FH] that in the associated graded algebras the structure constants with respect to the basis of conjugacy class sums are independent of $n$. This stability leads to the construction of a universal stable ring (the Farahat-Higman ring), equipped with a distinguished basis. Remarkably, this ring can be identified with the ring of symmetric functions but endowed with a new basis, see [Ma].

The above stability result has been generalized by Wang [Wa] to wreath products $\Gamma\wr S_{n}$ for any finite group $\Gamma$ and moreover it is shown in [Wa] that when the group $\Gamma$ is a finite subgroup of $SL_{2}(\mathbb{C})$, the associated graded algebra of the center of the group algebra of the wreath product is isomorphic to the cohomology ring of Hilbert scheme of $n$ points on the minimal resolution of $\mathbb{C}^{2}/\Gamma$. Recently the first author and Wang [WW] generalized the above stability property to the general linear group $GL_{n}(q)$ over a finite field $\mathbb{F}_{q}$. More precisely, the group $GL_{n}(q)$ is proved to be generated by the reflections in $GL_{n}(q)$ and hence the center of the integral group algebra $\mathbb{Z}[GL_{n}(q)]$ admits a filtration with respect to the reflection length. Then the structure constants of the associated graded algebras are shown to be independent of $n$, and this stability leads to a universal stable center with positive integer structure constants which governs the graded algebras for all $n$.

Based on [WW], Özden established the stability property for the family of symplectic groups $Sp_{2n}(q)$ in [Oz] with respect to length function in terms of reflections in $GL_{2n}(q)$. Later on, Kannan-Ryba [KR] proved that the structure constants of the centers of the integral group algebra of the classical finite groups over $\mathbb{F}_{q}$ are polynomials in $q^{n}$, recovering the results in [Me] for the case of $GL_{n}(q)$.

Besides the classical finite groups, the general affine groups $GA_{n}(q)$ over ${\mathbb{F}}_{q}$ form another rich and sophisticated family of finite groups which has been initially studied by Zelevinsky [Ze]. The main goal of this paper is to formulate and establish a stability result à la Farahat-Higman for the centers of the integral group algebras of $GA_{n}(q)$.

Denote by $\Phi_{q}$ the set of monic irreducible polynomials in ${\mathbb{F}}_{q}[t]$ other than $t$. It is well known (cf. [Ma]) that the conjugacy classes of $GL_{n}(q)$ are parametrized by the types ${\boldsymbol{\lambda}}=({\boldsymbol{\lambda}}(f))_{f\in\Phi_{q}}\in\mathscr{P}_{n}(\Phi_{q})$ (which are the partition-valued functions on $\Phi_{q}$ of degree $n$; cf. (2.2)). The general affine group $GA_{n}(q)$ is the subgroup of $GL_{n}(q)$ consisting of matrices of the form $\begin{bmatrix}1&0\\ \alpha&g\end{bmatrix}$ with $g\in GL_{n-1}(q)$ and $\alpha\in{\mathbb{F}}_{q}^{n-1}$. The conjugacy classes of $GA_{n}(q)$ turns out to be parametrized by the set of pair $({\boldsymbol{\lambda}},k)$, where ${\boldsymbol{\lambda}}\in\mathscr{P}_{n-1}(\Phi_{q})$ and $k\in\mathbb{N}$ satisfying $m_{k}\geq 1$ if $k\geq 1$, where ${\boldsymbol{\lambda}}(t-1)$ is written as ${\boldsymbol{\lambda}}(t-1)=(1^{m_{1}}2^{m_{2}}\cdots)$. An element $A=\begin{bmatrix}1&0\\ \alpha&g\end{bmatrix}\in GA_{n}(q)$ belongs to the conjugacy class corresponding to $({\boldsymbol{\lambda}},k)$ with $g\in GL_{n-1}(q)$ of type ${\boldsymbol{\lambda}}\in\mathscr{P}_{n-1}(\Phi_{q})$ and $k\geq 0$ uniquely determined by $A$ and accordingly $A$ is said to be of type $({\boldsymbol{\lambda}},k)$. Furthermore, we define the modified type of $A$ to be $(\mathring{{\boldsymbol{\lambda}}},k)$, where $\mathring{{\boldsymbol{\lambda}}}$ is the modified type of $g$ introduced in [WW], that is, $\mathring{{\boldsymbol{\lambda}}}(f)={\boldsymbol{\lambda}}(f)$ for $f\neq t-1$ and $\mathring{{\boldsymbol{\lambda}}}(t-1)=({\boldsymbol{\lambda}}^{e}_{1}-1,{\boldsymbol{\lambda}}^{e}_{2}-1,\ldots,{\boldsymbol{\lambda}}^{e}_{r}-1)$ if ${\boldsymbol{\lambda}}(t-1)=({\boldsymbol{\lambda}}^{e}_{1},{\boldsymbol{\lambda}}^{e}_{2},\ldots,{\boldsymbol{\lambda}}^{e}_{r})$. he key property is that the modified type remains unchanged for $A$ under the embedding of $GA_{n}(q)$ into $GA_{n+1}(q)$ and it is also clearly conjugation invariant. It follows that the conjugacy classes of $GA_{\infty}(q)=\cup_{n\geq 1}GA_{n}(q)$ are parametrized by the modified types in $\widehat{\mathscr{P}}^{\mathsf{a}}(\Phi_{q})$, see (2.31) for the notation. Let $\mathscr{K}_{({\boldsymbol{\lambda}},k)}(n)\text{ be }\text{conjugacy class of }GA_{n}(q)\text{ of modified type }({\boldsymbol{\lambda}},k)$. Denote ${P}_{({\boldsymbol{\lambda}},k)}(n)$ the class sum of elements in $GA_{n}(q)$ of modified type $({\boldsymbol{\lambda}},k)$, and write

with $\mathfrak{p}_{({\boldsymbol{\lambda}},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)$ being the structure constants.

It is known that the set of reflections (which are the elements with fixed point subspace in ${\mathbb{F}}_{q}^{n}$ having codimension one) in $GL_{n}(q)$ forms a generating set for $GL_{n}(q)$ and the reflection length $\ell(g)$ of a general element $g\in GL_{n}(q)$ is by definition the length of any reduced word of $g$ in terms of reflections. It turns out that the affine group $GA_{n}(q)\subset GL_{n}(q)$ is generated by the set of reflections belonging to the affine group $GA_{n}(q)$. Accordingly, the length $\ell^{\mathsf{a}}(A)$ of a general element $A\in GA_{n}(q)$ is by definition the length of any reduced word of $A$ in terms of reflections in $GA_{n}(q)$; two conjugate elements in $GA_{n}(q)$ have the same length. It turns out that $\ell^{\mathsf{a}}(A)=\|({\boldsymbol{\lambda}},k)\|$ if $A$ is of modified type $({\boldsymbol{\lambda}},k)$, see (3.15) for notations. The center $\mathcal{A}_{n}(q)$ of the integral group algebra $\mathbb{Z}[GA_{n}(q)]$ of $GA_{n}(q)$ is a filtered algebra with a basis of conjugacy class sums with respect to the reflection length. Denote by $\mathcal{G}_{n}(q)$ the associated graded algebra. Then we have

Our main result concerning the stability of the structure constants $\mathfrak{p}_{({\boldsymbol{\lambda}},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)$ is as follows.

The proof of Theorem 3.11 relies on two key observations. One observation is that the series $GA_{1}(q)\subset GA_{2}(q)\subset\cdots\subset GA_{n}(q)\subset GA_{n+1}(q)\subset\cdots$ satisfies the so-called strictly increasing property, that is, $\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)\leq\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n+1)$ for any admissible $n\geq 1$ and moreover if $\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)<\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n+1)$ then $\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n+1)<\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n+2)$. Another key observation is that $\ell^{a}(A)$ coincides with $\ell(A)$ for any $A\in GA_{n}(q)$ and hence when $\|(\boldsymbol{{\boldsymbol{\nu}}},t)\|=\|({\boldsymbol{\lambda}},k)\|+\|(\boldsymbol{{\boldsymbol{\mu}}},s)\|$ the increasing sequence $\cdots\leq\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)\leq\mathfrak{p}_{(\boldsymbol{\lambda},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n+1)\leq\cdots$ is bounded by a constant independent of $n$ by applying the stability property established in [WW].

Theorem 3.11 can be rephrased as that the associated graded algebra $\mathcal{G}_{n}(q)$ of $\mathcal{A}_{n}(q)$ has structure constants independent of $n$. We introduce a graded $\mathbb{Z}$-algebra $\mathcal{G}(q)$ with a basis given by the symbols ${P}_{({\boldsymbol{\lambda}},k)}$ indexed by $({\boldsymbol{\lambda}},k)\in\widehat{\mathscr{P}}^{\mathsf{a}}(\Phi_{q})$, and its multiplication has structure constants $\mathfrak{p}^{({\boldsymbol{\nu}},t)}_{({\boldsymbol{\lambda}},k),({\boldsymbol{\mu}},s)}$ as in the theorem above, for $\|({\boldsymbol{\nu}},t)\|=\|({\boldsymbol{\lambda}},k)\|+\|({\boldsymbol{\mu}},s)\|$; cf. (3.20).

We compute some examples of the structure constants $\mathfrak{p}_{({\boldsymbol{\lambda}},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)$ in $\mathcal{A}_{n}(q)$. Our computation turns out to imply two interesting facts. One fact is that we show the structure constant $a_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}$ in the center of $\mathbb{Z}[GL_{n}(q)]$ in the case $\|{\boldsymbol{\nu}}\|=\|{\boldsymbol{\lambda}}\|+\|\boldsymbol{{\boldsymbol{\mu}}}\|$ coincides with $\mathfrak{p}_{({{\boldsymbol{\lambda}}},0),({\boldsymbol{\mu}},0)}^{({\boldsymbol{\nu}},0)}$. This means the structure constants in the graded algebra for $GL_{n}(q)$ studied in [WW] are special cases of the structure constants in the graded algebra $\mathcal{G}_{n}(q)$ for $GA_{n}(q)$.

It is known [Be, DL] that in $GA_{n}(q)$ there exists another set of generators consisting of affine reflections and moreover the affine reflection length denoted by $\ell\ell^{\mathsf{a}}(A)$ satisfies $\ell\ell^{\mathsf{a}}(A)=\ell\ell^{\mathsf{a}}(B)$ whenever $A,B\in GA_{n}(q)$ are conjugate. Hence the center $\mathcal{A}_{n}(q)$ of the integral group algebra $\mathbb{Z}[GA_{n}(q)]$ of $GA_{n}(q)$ admits another filtered structure with respect to the affine reflection length. It turns out that the structure constants of the corresponding graded algebra $\mathcal{G}^{\prime}_{n}(q)$ actually dependents on $n$, which is different from $\mathcal{G}_{n}(q)$. This difference is an interesting phenomenon which is worthwhile to be further explored to get deeper connection between these two algebras. It also indicate that the algebra $\mathcal{G}_{n}(q)$ defined via the length function $\ell^{\mathsf{a}}(A)$ is the suitable one for further study.

The paper is organized as follows. In Section 2, we review basic facts on $GA_{n}(q)$ and in particularly give an explicit representatives for each conjugacy class of $GA_{n}(q)$. We introduce the notion of modified types for each element in $GA_{n}(q)$. In Section 3, we recall the stability property for $GL_{n}(q)$ established in [WW] and prove that the center $\mathcal{A}_{n}(q)$ of $\mathbb{Z}[GA_{n}(q)]$ is a filtered algebra. Then we formulate and establish the stability on the structure constants for the graded algebra $\mathcal{G}_{n}(q)$ and the universal stable center $\mathcal{G}(q)$. In Section 4, we compute some examples of the structure constants $\mathfrak{p}_{({\boldsymbol{\lambda}},k),({\boldsymbol{\mu}},s)}^{({\boldsymbol{\nu}},t)}(n)$ and prove the structure constants $a_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}$ in the situation $\|\boldsymbol{{\boldsymbol{\nu}}}\|=\|{\boldsymbol{\lambda}}\|+\|\boldsymbol{{\boldsymbol{\mu}}}\|$ in the center of $\mathbb{Z}[GL_{n}(q)]$ coincides with $\mathfrak{p}_{({\boldsymbol{\lambda}},0),({\boldsymbol{\mu}},0)}^{({\boldsymbol{\nu}},0)}$.

Acknowledgements. This project was partially carried out while the first author enjoyed the support and hospitality of University of Virginia. She would like to thank Weiqiang Wang and Arun Kannan for many helpful discussions on the project. Both author are supported by NSFC-12122101 and NSFC-12071026.

Let $\mathscr{P}$ be the set of all partitions. For $\lambda=(\lambda_{1},\lambda_{2},\ldots,)\in\mathscr{P}$, we denote its size by $|\lambda|=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{\ell}$, its length by $\ell(\lambda)$. We will also write $\lambda=(1^{m_{1}(\lambda)}2^{m_{2}(\lambda)}\ldots)$, where $m_{i}(\lambda)$ is the number of parts in $\lambda$ equal to $i$. For two partitions $\lambda,\mu\in\mathscr{P}$, we denote by $\lambda\cup\mu=(1^{m_{1}(\lambda)+m_{1}(\mu)}2^{m_{2}(\lambda)+m_{2}(\mu)}\cdots)$ the partition whose parts are those of $\lambda$ and $\mu$. For a set $Y$, let $\mathscr{P}(Y)$ be the set of the partition-valued functions $\boldsymbol{\lambda}:Y\rightarrow\mathscr{P}$ such that only finitely many ${\boldsymbol{\lambda}}(y)$ are nonempty partitions. Given ${\boldsymbol{\lambda}},{\boldsymbol{\mu}}\in\mathscr{P}(Y)$, we define ${\boldsymbol{\lambda}}\cup{\boldsymbol{\mu}}\in\mathscr{P}(Y)$ by letting $({\boldsymbol{\lambda}}\cup{\boldsymbol{\mu}})(y)={\boldsymbol{\lambda}}(y)\cup{\boldsymbol{\mu}}(y)$ for each $y\in Y$.

Denote by $\mathbb{F}_{q}$ the finite field of $q$ elements, where $q$ is a prime power. We shall regard vectors in the $n$-dimensional vector space ${\mathbb{F}}_{q}^{n}$ as column vectors, that is, ${\mathbb{F}}_{q}^{n}=\{v=(v_{1},\ldots,v_{n})^{\intercal}|v_{k}\in{\mathbb{F}}_{q},1\leq k\leq n\}$ for each $n\geq 1$. Denote by $M_{n\times m}(q)$ the set of $n\times m$ matrices over the finite field $\mathbb{F}_{q}$. The general linear group $GL_{n}(q)$, which consists of all invertible matrices in $M_{n\times n}(q)$, acts on $\mathbb{F}_{q}^{n}$ naturally via left multiplication.

Following [Ma], we recall the description of conjugacy classes in $GL_{n}(q)$. Let $\Phi_{q}$ be the set of all monic irreducible polynomial in ${\mathbb{F}}_{q}[t]$ other than $t$. Then for each $g\in GL_{n}(q)$, the vector space ${\mathbb{F}}_{q}^{n}$ admits a ${\mathbb{F}}_{q}[t]$-module via $t\cdot v=gv$ for any $v\in{\mathbb{F}}_{q}^{n}$, Since ${\mathbb{F}}_{q}[t]$ is a PID, there exists a unique ${\boldsymbol{\lambda}}=({\boldsymbol{\lambda}}(f))_{f\in\Phi_{q}}\in\mathscr{P}(\Phi_{q})$ such that

where we write ${\boldsymbol{\lambda}}(f)=({\boldsymbol{\lambda}}_{1}(f),{\boldsymbol{\lambda}}_{2}(f),\ldots)\in\mathscr{P}$; moreover, we have

where $d(f)$ denotes the degree of the polynomial $f$. Denote by $\mathscr{P}_{n}(\Phi_{q})$ the set of ${\boldsymbol{\lambda}}\in\mathscr{P}(\Phi_{q})$ satisfying (2.2). The partition-valued function ${\boldsymbol{\lambda}}=({\boldsymbol{\lambda}}(f))_{f\in\Phi_{q}}\in\mathscr{P}(\Phi_{q})$ is called the type of $g$. Then any two elements of $GL_{n}(q)$ are conjugate if and only if they have the same type, and there is a bijection between the set of conjugacy classes of $GL_{n}(q)$ and the set $\mathscr{P}_{n}(\Phi_{q})$. For each ${\boldsymbol{\lambda}}\in\mathscr{P}_{n}(\Phi_{q})$, denote by $\mathcal{K}_{\boldsymbol{\lambda}}$ the conjugacy class of elements in $G_{n}$ of type ${\boldsymbol{\lambda}}$.

For each $f=t^{d}-\sum_{1\leq i\leq d}a_{i}t^{i-1}\in\Phi_{q}$, let $J(f)$ denote the companion matrix for $f$ of the form

and for each integer $m\geq 1$ let

with $m$ diagonal blocks $J(f)$, where $I_{d}$ is the $d\times d$ identity matrix. Given ${\boldsymbol{\lambda}}\in\mathscr{P}(\Phi_{q})$ with ${\boldsymbol{\lambda}}(f)=({\boldsymbol{\lambda}}_{1}(f),{\boldsymbol{\lambda}}_{2}(f),\ldots)$, set

that is, $J_{\boldsymbol{\lambda}}$ is the diagonal sum of the matrices $J_{{\boldsymbol{\lambda}}_{i}(f)}(f)$ for all $i\geq 1$ and $f\in\Phi_{q}$. Then an element $g\in GL_{n}(q)$ of type ${\boldsymbol{\lambda}}$ is conjugate to the canonical form $J_{\boldsymbol{\lambda}}$ (cf. [Ma, Chapter IV, §2]). For $f\in\Phi_{q}$, set

Then by (2.1), we have $V_{J_{\boldsymbol{\lambda}}(f)}\cong\oplus_{i\geq 1}{\mathbb{F}}_{q}[t]/(f)^{{\boldsymbol{\lambda}}_{i}(f)}$ as ${\mathbb{F}}_{q}[t]$-modules and moreover, we have $J_{\boldsymbol{\lambda}}=\text{diag}\,\big{(}J_{\boldsymbol{\lambda}}(f)\big{)}_{f\in\Phi_{q}}.$ Set

Suppose ${\boldsymbol{\lambda}}(t-1)=(1^{m_{1}}2^{m_{2}}\cdots r^{m_{r}})$ and set

for $1\leq k\leq r$, then we can write

By [WW, Lemma 2.1] (cf. [Ma, IV, (2.5)]), we have the following.

Recall $J_{m}(t-1)$ is the Jordan form of size $m$ and eigenvalue $1$. The following elementary lemma can be verified by a direct computation.

The following lemma will be useful to compute the conjugacy classes in $GA_{n}(q)$.

We will review some basics on affine spaces and general affine group by following [DL]. Let $V$ be a finite dimensional vector space over $\mathbb{F}_{q}$. An affine space $\widetilde{V}$ associated to $V$ is defined to be a set $\widetilde{V}$ together with a unique $V$-action $V\times\widetilde{V}\rightarrow\widetilde{V},(\alpha,x)\mapsto\alpha+x$ satisfies

for any $x\in\widetilde{V},\alpha,\beta\in V$ and moreover for any $x,y\in\widetilde{V}$, there exists a unique vector $\alpha\in V$ such that $\alpha+x=y$. For $x,y\in\widetilde{V}$, as there exists a unique vector $\alpha\in V$ such that $\alpha+x=y$, we can define the difference $y-x=\alpha$ which is a vector in $V$. However, one cannot take sums of elements of $\widetilde{V}$. We define $\dim\widetilde{V}=\dim V$. A map $f:\widetilde{V}\rightarrow\widetilde{V}$ is said to be an affine transformation if

for any $x_{1},x_{2},y_{1},y_{2}\in\widetilde{V}$ satisfying $y_{1}-x_{1}=y_{2}-x_{2}\in V$ and moreover the induced map $f^{0}:V\rightarrow V,\alpha\mapsto f(\alpha+x)-f(x)$ is linear, where $x\in\widetilde{V}$ is a fixed point. We observe that the map $f^{0}$ is well-defined due to (2.9). Clearly the composition of two affine transformations is still an affine transformation. An affine transformation $f$ is called invertible if it is bijective and it is straightforward to check that the inverse of an invertible affine transformation is still an affine transformation. The (general) affine group $GA(\widetilde{V})$ of $\widetilde{V}$ is defined to be group consisting of all invertible affine transformations of $\widetilde{V}$.

In an affine space, it is possible to fix a point and coordinate axis such that every point in the space can be represented as a $n$-tuple of its coordinates. For example, suppose $\dim\widetilde{V}=\dim V=n-1$. By choosing coordinates for $\widetilde{V}$, one can identify $\widetilde{V}$ with the affine hyperplane

in $\mathbb{F}_{q}^{n}$ and $V$ is identified with the linear subspace $\{(0,v_{1},v_{2},\ldots,v_{n-1})^{\intercal}|v_{1},\ldots,v_{n-1}\in\mathbb{F}_{q}\}$. Then general affine group $GA(\widetilde{V})$ can be identified with the set of $n\times n$-matrices:

It is known [Ze] that the number of conjugacy classes of $GA_{n}(q)$ is equal to $c_{0}+c_{1}+\cdots+c_{n-1}$ with $c_{i}$ being the number of conjugacy classes of $GL_{i}(q)$ and a representative of conjugacy classes is also given in [Mu] via a “maximal problem” procedure in the context of maximal parabolic subgroups. In the following, we shall give an explicit description of the type of each matrix in $GA_{n}(q)$ as well as an explicit representative for each conjugacy class.

Recall the general affine group $GA_{n}(q)$ is the subgroup of $GL_{n}(q)$ given by

Clearly the natural embedding $GL_{n}(q)\subset GL_{n+1}(q),h\mapsto\begin{bmatrix}h&0\\ 0&1\end{bmatrix}$ for $n\geq 0$ leads to the following embedding and hence:

Denote by $GA_{\infty}(q)=\cup_{n\geq 1}GA_{n}(q)$ the corresponding limit group. For simplicity, write $\text{Im}(I_{n-1}-g)=\{(I_{n-1}-g)\beta|\beta\in{\mathbb{F}}_{q}^{n-1}\}$ for each $g\in GL_{n-1}(q)$. The following formula will be useful later for our computation:

for any $A=\begin{bmatrix}1&0\\ \alpha&g\end{bmatrix},B=\begin{bmatrix}1&0\\ \beta&h\end{bmatrix}\in GA_{n}(q)$.