Stability for Representations of Hecke Algebras of Type A

Abstract

In this paper, we introduce the concept of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

1. Introduction

Representation theory uses a class of matrices to represent abstract algebraic objects and makes the operations in this original structure correspond to matrix operations. This method can be applied to many kinds of algebraic structures, such as group, associative algebra, and Lie algebra. Recently, inspired by the work of Church, Ellenberg, Farb, and Nagpal in References [1,2], people have been studying many problems in basic mathematics, such as algebra, algebraic geometry, and number theory, through the theory of $FI$-modules, where $FI$ is the category of finite sets and injective maps. Essentially, $FI$-module over a commutative ring R is a functor from the category $FI$ to the category of R-modules. Thus, a sequence of representations of symmetric groups can be described by a functor between categories, that is, an $FI$-module. Many important results have been obtained, such as homological properties of $FI$-modules, Noetherian properties, and representation stability properties, in a series of articles [3,4,5,6,7,8].

Let $G_n$ be one of the symmetric group ${\mathfrak{S}}_n$, the general linear group $G{L}_n(\mathbb{Q})$, or the special linear group $S{L}_n(\mathbb{Q})$. In 2013, Church and Farb introduced the theory of representation stability to study the representations $V_n$ of these natural groups $G_n$ [9]. Let us review some basic concepts before introducing relevant background. The idea is mainly to study the consistent sequence of representations $V_n$ of groups $G_n$ as follows. A sequence of $G_n$-representations,

$$V_0\stackrel{{\varphi }_0}{⟶}{V}_1\stackrel{{\varphi }_1}{⟶}{V}_2\stackrel{{\varphi }_2}{⟶}\cdots ,$$

is called consistent sequence if, for all $n\ge 0$ and $g\in G_n$, the diagram

$$\begin{array}{ccc}{V}_n& \stackrel{{\varphi }_n}{\to }& V_{n+1}\\ g\downarrow \phantom{g}& & g\downarrow \phantom{g}& & \\ V_n& \stackrel{{\varphi }_n}{\to }& V_{n+1}\end{array}$$

is commutative, where each ${\varphi }_n$ is a linear map.

A consistent sequence is called representation stable if, for sufficiently large n, it satisfies

• Injectivity: The map ${\varphi }_n$ is injective;
• Surjectivity: The span of the image of ${\varphi }_n$ is equal to $V_{n+1}$;
• Multiplicities: Decompose $V_n$ into irreducible representations as

  $$V_n=\underset{\lambda }{⨁}{c}_{\lambda ,n}V{(\lambda )}_n,$$

  with multiplicities $0\le c_{\lambda ,n}\le \infty$. For each partition $\lambda$, $c_{\lambda ,n}$ is independent of n.

Church, Ellenberg, and Farb used the theory of $FI$-modules to study the consistent sequence of representations of the symmetric groups ${\mathfrak{S}}_n$ over a commutative ring R [1]. It has been proved that, if a consistent sequence of representations of symmetric group over a field of characteristic zero as $FI$-module is finitely generated, then this consistent sequence is representation stable [1].

In Reference [2], Church, Ellenberg, Farb, and Nagpal proved the Noetherian property for $FI$-modules over Noetherian rings. The consistent sequence of representations of symmetric groups given by a finitely generated $FI$-module turned out to be representation stable.

In Reference [10], Gan and Watterlond researched $VI$-modules, where $VI$ is the category of finite dimensional vector spaces and injective linear maps. They proved a result about the representation stability for the family of finite general linear groups $G{L}_n({\mathbb{F}}_q)$, where q is a power of prime. That is, a $VI$-module over an algebraically closed field of characteristic zero is finitely generated if and only if the consistent sequence obtained from V is representation stable and $dim(V_n)<\infty$ for each n. In Reference [11], d’Andecy and Walker introduced the cyclotomic quotients of affine Hecke algebras of type D. They studied the finite dimensional representations of affine Hecke algebras of type D through the representation theory of some cyclotomic quotients. In Reference [12], Laudone investigated a similar category with $FI$ for the 0-Hecke algebra ${\mathcal{H}}_n(0)$ called the 0-Hecke category. They obtained a new type of representation stability in this setting and proved that it is implied by a finitely generated $\mathcal{H}$-module.

Hecke algebra ${\mathcal{H}}_n$ of type A is the deformation of group algebra $\mathbb{C}[{\mathfrak{S}}_n]$, and the representations of ${\mathcal{H}}_n$ is also the deformation of representations of ${\mathfrak{S}}_n$. In the light of works of Church, Ellenberg, and Farb, the relation between the representation categories of $\mathbb{C}[{\mathfrak{S}}_n]$ and ${\mathcal{H}}_n$ strongly suggests the existence of representation stability of Hecke algebras ${\mathcal{H}}_n$. This is the original motivation of this paper. The rest of the paper is organized as follows. Firstly, we review some basic concepts of Hecke algebras, partitions, and give the definition of stability for representations of Hecke algebras. Secondly, we discuss the injective degree, surjective degree, stability degree, weight, and Noetherian property of $F{I}_{\mathcal{H}}$-modules. Finally, we give the main theorem of this paper. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

2. Preliminaries

Let us first list some basic results about Hecke algebras and some notations.

Let k be a field of characteristic 0. $q\in k$ is not a root of unity. Let $n\ge 2$ and ${\mathfrak{S}}_n$ be the symmetric group. It is easy to show that ${\mathfrak{S}}_n$ is generated by the $n-1$ transpositions $\{(1,2),(2,3),\cdots ,(n-1,n)\}$. For the convenience of writing, we denote the transposition $(i,i+1)$ by $s_i$ for $1\le i\le n-1$. The Iwahori-Hecke algebra (or simply the Hecke algebra) ${\mathcal{H}}_n={\mathcal{H}}_{k,q}({\mathfrak{S}}_n)$ of the symmetric group ${\mathfrak{S}}_n$ is the unital associative k-algebra with generators $\{T_{s_1},T_{s_2},\cdots ,T_{s_{n-1}}\}$ and relations

$(T_{s_i}-q)(T_{s_i}+1)=0$, for $i=1,2,\cdots ,n-1$;

$T_{s_i}{T}_{s_j}=T_{s_j}{T}_{s_i}$, for $1\le i<j-1\le n-2$;

$T_{s_i}{T}_{s_{i+1}}{T}_{s_i}=T_{s_{i+1}}{T}_{s_i}{T}_{s_{i+1}}$, for $i=1,2,\cdots ,n-2$.

We shall denote ${\mathcal{H}}_0={\mathcal{H}}_1=k$.

Suppose $\sigma \in {\mathfrak{S}}_n$. Then, $\sigma$ can be expressed as the product of some transpositions. The expression that uses the minimum number of transpositions is called the reduced expression. This number about $\sigma$ is denoted by $l(\sigma )$; thus, we have

$$l(\sigma )=min\{m|\sigma =s_{i_1}\cdots s_{i_m}\}.$$

Denote $T_{\sigma }:=T_{s_{i_1}}\cdots T_{s_{i_l}}$ which is an element of the Hecke algebra ${\mathcal{H}}_n$. Then, ${\mathcal{H}}_n$ is free with k-basis $\{T_{\sigma }|\sigma \in {\mathfrak{S}}_n\}$. If ${\sigma }_1\in {\mathfrak{S}}_n,{\sigma }_2\in {\mathfrak{S}}_n$ and $l({\sigma }_1{\sigma }_2)=l({\sigma }_1)+l({\sigma }_2)$, then it is not difficult to obtain that $T_{{\sigma }_1{\sigma }_2}=T_{{\sigma }_1}{T}_{{\sigma }_2}$.

We now review the concept of composition (partition) of natural numbers, which will be used in later sections. Let $n\in \mathbb{N}$. A sequence of non-negative integers $\lambda =({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m)$ which satisfies ${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_m=n$ is called a composition of n. A composition $\lambda$ is called a partition of n when ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_m\ge 0$. Composition (partition) $\lambda$ of n is generally denoted by $\lambda \vDash n$ ($\lambda ⊢n$). Let $1\le k,l\le n$ and $k+l\le n$. Obviously, the set

$$S_{\{k+1,k+2,\cdots ,k+l\}}\triangleq \left\{\sigma \in {\mathfrak{S}}_n|\sigma (i)=i,i\in \{1,2,\cdots ,n\}-\{k+1,k+2,\cdots ,k+l\}\right\}$$

is a subgroup of ${\mathfrak{S}}_n$. For each composition $\lambda$ of n, the standard Young subgroup ${\mathfrak{S}}_{\lambda }$ of ${\mathfrak{S}}_n$ is defined by the direct product $S_{\{1,2,\cdots ,{\lambda }_1\}}\times S_{\{{\lambda }_1+1,{\lambda }_1+2,\cdots ,{\lambda }_1+{\lambda }_2\}}\times \cdots \times$$S_{\{{\lambda }_1+{\lambda }_2+\cdots +{\lambda }_{n-1}+1,\cdots ,n\}}$. For example, if $\lambda =(3,2,1)$ is a composition of 6, then the corresponding Young subgroup is ${\mathfrak{S}}_3\times \{(1),(4,5)\}$. For more details of Young subgroups, please refer to Reference [13]. The subalgebra ${\mathcal{H}}_{\lambda }$ of $\mathcal{H}$ generated by $\{T_{\sigma }|\sigma \in {\mathfrak{S}}_{\lambda }\}$ is k-free with basis $\{T_{\sigma }|\sigma \in {\mathfrak{S}}_{\lambda }\}$. The subalgebras of these types are the so-called Young subalgebras.

Let $\lambda =({\lambda }_1,{\lambda }_2,\cdots )$ be a partition of some natural number. For any $n\ge \left|\lambda \right|+{\lambda }_1$, the padded partition is defined as

$$\lambda [n]:=(n-|\lambda |,{\lambda }_1,{\lambda }_2,\cdots ),$$

where $\left|\lambda \right|={\displaystyle \sum _{i=1}^{\infty }}{\lambda }_i$.

In the case of the Hecke algebra ${\mathcal{H}}_n$ that we have assumed, the simple module of Hecke algebra can be labeled (up to isomorphism) by the set of all partitions of n. For a partition $\lambda$ and $n\ge \left|\lambda \right|+{\lambda }_1$, we denote $S^{\lambda }$ the corresponding simple ${\mathcal{H}}_{\left|\lambda \right|}$-module and set $S{(\lambda )}_n:=S^{\lambda [n]}$.

Since ${\mathfrak{S}}_n$ is naturally regarded as a subgroup of ${\mathfrak{S}}_{n+1}$, then the generators of Hecke algebra ${\mathcal{H}}_n$ is a subset of generators of Hecke algebra ${\mathcal{H}}_{n+1}$. That is to say, ${\mathcal{H}}_n$ is a subalgebra of ${\mathcal{H}}_{n+1}$. The natural injective k-algebra map from ${\mathcal{H}}_n$ to ${\mathcal{H}}_{n+1}$ is denoted by $\tau$.

Definition 1.

A sequence $V={(V_n,{\varphi }_n)}_{n\in \mathbb{N}}$ is called consistent sequence or $F{I}_{\mathcal{H}}$-module if it satisfies: for all $h\in {\mathcal{H}}_n$, the diagram

$$\begin{array}{ccc}{V}_n& \stackrel{{\varphi }_n}{\to }& V_{n+1}\\ h\downarrow \phantom{h}& & \tau (h)\downarrow \phantom{\tau (h)}& & \\ V_n& \stackrel{{\varphi }_n}{\to }& V_{n+1}\end{array}$$

is commutative, where $V_n$ is ${\mathcal{H}}_n$-module, and ${\varphi }_n:V_n\to V_{n+1}$ is k-linear map.

Let $V=(V_n,{\varphi }_n)$ and $W=(W_n,{\psi }_n)$ be two $F{I}_{\mathcal{H}}$-modules. A morphism $f=(f_n)$ from V to W is a sequence of homomorphisms $f_n:V_n\to W_n$ for all $n\ge 0$ such that the following diagram

$$\begin{array}{ccc}{V}_n& \stackrel{{\varphi }_n}{\to }& V_{n+1}\\ f_n\downarrow \phantom{f_n}& & f_{n+1}\downarrow \phantom{f_{n+1}}& & \\ W_n& \stackrel{{\psi }_n}{\to }& W_{n+1}\end{array}$$

is commutative.

Remark 1.

$F{I}_{\mathcal{H}}$-modules, together with the above morphisms, are an abelian category. Notions, such as kernel, cokernel, injection, and surjection, are defined by pointwise. The direct sum and tensor product also make sense. For example, let $f=(f_n):V\to W$ be a morphism. We need to find an $F{I}_{\mathcal{H}}$-module, and the kernel of this morphism satisfies the kernel’s universal property of this category. As usual, this $F{I}_{\mathcal{H}}$-module is denoted by $kerf$. In addition, the construction of the $F{I}_{\mathcal{H}}$-module $kerf$ is through with defining at each point n. That is, ${(kerf)}_n$ is defined by $ker(f_n)$, and the map from ${(kerf)}_n$ to ${(kerf)}_{n+1}$ is the restriction of ${\varphi }_n$. Now, it is easy to check that, for all $h\in {\mathcal{H}}_n$, we have commutative composite mappings, i.e., $\tau (h){\varphi }_n{{|}_{ker(f_n)}={\varphi }_n|}_{ker(f_n)}h$, which makes $kerf$ an $F{I}_{\mathcal{H}}$-module. Similarly, we can also derive the tensor product of two $F{I}_{\mathcal{H}}$-modules straightforward. The tensor product $V⨂W$ of two $F{I}_{\mathcal{H}}$-modules $V=(V_n,{\varphi }_n)$ and $W=(W_n,{\psi }_n)$ is defined by ${(V⨂W)}_n=V_n{⨂}_k{W}_n$, and the map from ${(V⨂W)}_n$ to ${(V⨂W)}_{n+1}$ is defined by ${\varphi }_n\otimes {\psi }_n$.

Definition 2.

Let $W={(W_n)}_{n\ge 0}$, where $W_n$ is an ${\mathcal{H}}_n$-module. Denote ${\lambda }_{m,n}$ the composition $(m,n-m)$ for $m\le n$. Then, $W_m$ will be an ${\mathcal{H}}_{{\lambda }_{m,n}}$-module under the action $T_{\sigma }w=T_{{\sigma }_1{\sigma }_2}w=T_{{\sigma }_1}w$, where $\sigma ={\sigma }_1{\sigma }_2$ for ${\sigma }_1\in S_{\{1,2,\dots ,m\}}$ and ${\sigma }_2\in S_{\{m+1,\dots ,n-m\}}$ and $w\in W_m$. We define a sequence $M(W)$ for $n\ge 0$,

$$M{(W)}_n:=\underset{m\le n}{⨁}({\mathcal{H}}_n\underset{{\mathcal{H}}_{{\lambda }_{m,n}}}{⨂}{W}_m),$$

with the linear map ${\varphi }_n:M{(W)}_n\to W{(W)}_{n+1}$ determined by $h\otimes w_m\mapsto \tau (h)\otimes w_m$, where $h\in {\mathcal{H}}_n$ and $w_m\in W_m$.

It is easy to check that $M(W)$ is an $F{I}_{\mathcal{H}}$-module. If $W={(W_n)}_{n\ge 0}$ satisfies $W_m={\mathcal{H}}_m$ for some m and $W_n=0$ for $n\ne m$, then we denote $M(W)$ by $M(m)$.

Let $V=(V_n,{\varphi }_n)$ be an $F{I}_{\mathcal{H}}$-module. Denote ${\varphi }_{n-1,m}={\varphi }_{n-1}{\varphi }_{n-2}\cdots {\varphi }_m:V_m\to V_n$ for $0\le m\le n-2$ and ${\varphi }_{n-1,n-1}={\varphi }_{n-1}$. For a disjoint union, $S={\displaystyle \underset{n=0}{\overset{\infty }{⨆}}}{S}_n$, where $S_n\subseteq V_n$. Next, we define the concept of generation through the span of linear space. We define $span{(S)}_n$ to be the submodule of $V_n$ generated by $({\displaystyle \bigcup _{i=0}^{n-1}}{\varphi }_{n-1,i}(S_i))\cup S_n$. In addition, the map ${\tilde{\varphi }}_n:span{(S)}_n\to span{(S)}_{n+1}$ is the restriction of ${\varphi }_n$. It is trivial to check from the above discussion that $span(S)$ is an $F{I}_{\mathcal{H}}$-module.

Definition 3.

We say that an $F{I}_{\mathcal{H}}$-module V is generated in degree $\le n$ if V is generated by elements of $V_m$ for $m\le n$.

According to the above definition, if V is generated in degree $\le n$, then we have

$$span(\underset{m\le n}{⨆}{V}_m)=V.$$

Definition 4.

We say that an $F{I}_{\mathcal{H}}$-module V is finitely generated if there is a finite set S of the disjoint union $⨆V_n$ such that $span(S)=V$.

Definition 5 (Representation stability of consistent sequence).

Let ${(V_n,{\varphi }_n)}_{n\in \mathbb{N}}$ be an $F{I}_{\mathcal{H}}$-module. This sequence is called uniformly representation stable if there exists an integer N such that for each with $n\ge N$, the following conditions hold

• Injectivity: ${\varphi }_n:V_n\to V_{n+1}$ is injective;
• Surjectivity: $V_{n+1}$ is generated by the image of ${\varphi }_n$;
• Multiplicities: $V_n={⨁}_{\lambda }{c}_{\lambda ,n}S{(\lambda )}_n$, where the multiplicities $c_{\lambda ,n}$ is independent of n.

This definition of representation stability is in the sense of Church and Farb [9]. They introduced the idea of representation stability for a sequence of representations of groups. An important result in Reference [1] is that a consistent sequence of ${\mathfrak{S}}_n$-representations is representation stable can be converted to a finite generation property for an $FI$-module.

3. Stability Degree

In this section, we define the concepts of stability degree, injective degree and surjective degree for an $F{I}_{\mathcal{H}}$-module. For $W={(W_n)}_{n\ge 0}$, we will compute these numbers for the $F{I}_{\mathcal{H}}$-module $M(W)$. The following lemma shows that $M(W)$ is similar to the “free object”.

Lemma 1.

If an $F{I}_{\mathcal{H}}$-module $V=(V_n,{\varphi }_n)$ is generated in degree $\le d$, then there exists an epimorphism from ${\displaystyle \underset{i=0}{\overset{d}{⨁}}}M(V_i)$ to V.

Proof. 

For every $0\le n\le d$, $V_n$ is a summand of ${({\displaystyle \underset{i=0}{\overset{d}{⨁}}}M(V_i))}_n$ since $M{(V_n)}_n=V_n$ by definition. So, the functions from ${({\displaystyle \underset{i=0}{\overset{d}{⨁}}}M(V_i))}_n$ to $V_n$ are defined by projections. Now, let $n>d$, and then ${({\displaystyle \underset{i=0}{\overset{d}{⨁}}}M(V_i))}_n={\displaystyle \underset{i=0}{\overset{d}{⨁}}}({\mathcal{H}}_n{⨂}_{{\mathcal{H}}_{{\lambda }_i}}{V}_i)$. If $v_i\in V_i$ for $0\le i\le d$, then $f_n(1\otimes v_i)={\varphi }_{n-1}\cdots {\varphi }_{i+1}{\varphi }_i(v_i)$. The fact ${\varphi }_n{f}_n=f_{n+1}$ implies that $f={(f_n)}_{n\ge 0}$ is a morphism. From the construction of f and V generated in degree $\le d$, we know f is an epimorphism. □

Given $a\ge 0$ and an $F{I}_{\mathcal{H}}$-module $V=(V_n,{\varphi }_n)$, we define a sequence ${\Phi }_a(V)$ as follows. Denote by $Q_n$ the subspace of $V_{a+n}$ spanned by $\{T_{\sigma }v-v|\sigma \in S_{\{a+1,a+2,\dots ,a+n\}},v\in V_{a+n}\}$ for $n\ge 0$ and then define ${\Phi }_a{(V)}_n:=V_{a+n}/Q_n$. In addition, the map $T:{\Phi }_a{(V)}_n\to {\Phi }_a{(V)}_{n+1}$ is defined by $[v]\mapsto [{\varphi }_{a+n}(v)]$. The map T is well-defined since, for $T_{\sigma }v-v\in Q_n$, ${\varphi }_{a+n}(T_{\sigma }v-v)=T_{\sigma }{\varphi }_{a+n}(v)-{\varphi }_{a+n}(v)\in Q_{n+1}$.

Definition 6.

If there exists a natural number s such that for all $a\ge 0$, the map defined above is an isomorphism for $n\ge s$, then V is called to have stability degrees. If the map T is injective or surjective, then V is called to have injective degree s or surjective degree s. These facts are denoted by stab-deg$(V)$(or inj-deg$(V)$, sur-deg$(V)$)$=s$.

The following lemma is about double cosets of symmetric group of two certain Young subgroups. It is an interesting consequence of group theory. This result will be used in the proof of the next lemmas.

Lemma 2.

Given $a\ge 0$. For $n\ge 0$ and $m\le a+n$, let ${\lambda }_n=(m,a+n-m)$ and ${\mu }_n=(1,1,\dots ,1,n)$ be two compositions of $a+n$. Denote ${\mathcal{D}}_{{\mu }_n,{\lambda }_n}$ the representative elements of minimal length of the double cosets ${\mathfrak{S}}_{{\mu }_n}\backslash {\mathfrak{S}}_{a+n}/{\mathfrak{S}}_{{\lambda }_n}$. Then, ${\mathcal{D}}_{{\mu }_n,{\lambda }_n}\subseteq {\mathcal{D}}_{{\mu }_{n+1},{\lambda }_{n+1}}$ and if $n\ge m$, the set ${\mathcal{D}}_{{\mu }_n,{\lambda }_n}$ is independent of n.

Proof. 

By Lemma 1.7 in Reference [14], there is a bijection from the set of row-standard tableaux in $\mathcal{T}({\lambda }_n,{\mu }_n)$ to ${\mathcal{D}}_{{\mu }_n,{\lambda }_n}$. These row-stantard ${\lambda }_n$-tableaux of type ${\mu }_n$ are determined by the subset of $\{1,2,\dots ,a\}$ which is in the first row of ${\lambda }_n$. Denote the number of the set of row-standard tableaux in $\mathcal{T}({\lambda }_n,{\mu }_n)$ by ${\mathcal{T}}_n$. So, we have ${\mathcal{T}}_n\le {\mathcal{T}}_{n+1}$. If $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$ is the element of minimal length in a double coset ${\mathfrak{S}}_{{\mu }_n}\sigma {\mathfrak{S}}_{{\lambda }_n}$, then $\sigma$’s length is also minimal in the double coset ${\mathfrak{S}}_{{\mu }_{n+1}}\sigma {\mathfrak{S}}_{{\lambda }_{n+1}}$. We obtain that $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$ implies $\sigma \in {\mathcal{D}}_{{\mu }_{n+1},{\lambda }_{n+1}}$. If $n\ge m$, the number ${\mathcal{T}}_n$ is independent of n. So, we complete the proof of the lemma. □

For $\alpha \in {\mathfrak{S}}_{a+n}$, there exist $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$, $\beta \in {\mathfrak{S}}_{{\mu }_n}$, and $\gamma \in {\mathfrak{S}}_{{\lambda }_n}$ such that $\alpha =\beta \sigma \gamma$. According to the Lemma 1.6 in Reference [14], these elements can be chosen to satisfy $l(\alpha )=l(\beta )+l(\sigma )+l(\gamma )$. This property will be used in the following lemma.

Lemma 3.

Assume $W={(W_n)}_{n\ge 0}$, where $W_n$ is an ${\mathcal{H}}_n$-module for $n\ge 0$. Then, $M(W)$ has injective degree 0; if $W_i=0$ for all $i>m$, then we have surj-deg$(M(W))\le m$. Therefore, $M(m)$ has injective degree 0 and surjective degree m.

Proof. 

At first, we compute $M{(W)}_{a+n}$ by definition.

$$\begin{array}{cc}\hfill M{(W)}_{a+n}& =\underset{m\le a+n}{⨁}({\mathcal{H}}_{a+n}\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{W}_m)\hfill \\ \hfill & =\underset{m\le a+n}{⨁}(\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{W}_m)).\hfill \end{array}$$

The map ${\varphi }_{a+n}:M{(W)}_{a+n}\to M{(W)}_{a+n+1}$ is the direct sum of the maps: For $m\le a+n,\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$, ${\mathcal{H}}_{{\mu }_n}{T}_{\sigma }{⨂}_{{\mathcal{H}}_{{\lambda }_n}}{W}_m\to {\mathcal{H}}_{{\mu }_{n+1}}{T}_{\sigma }{⨂}_{{\mathcal{H}}_{{\lambda }_{n+1}}}{W}_m$ defined by $h{T}_{\sigma }\otimes w\mapsto \tau (h)T_{\sigma }\otimes w$, where $h\in {\mathcal{H}}_{{\mu }_n}$ and $w\in W_m$. In addition, so

$$\begin{array}{cc}\hfill {\Phi }_a{(M(W))}_n& =M{(W)}_{a+n}/Q_n\hfill \\ \hfill & \cong \underset{m\le a+n}{⨁}\left(\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}(({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{W}_m)/Q_n)\right)\hfill \\ \hfill & \cong \underset{m\le a+n}{⨁}(\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}(T_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{W}_m)).\hfill \end{array}$$

The map $T:{\Phi }_a{(M(W))}_n\to {\Phi }_a{(M(W))}_{n+1}$ restricted in one factor in the above decomposition is the identity. So, we have obtained that the inj-deg$(M(W))=0$. If $W_i=0$ for $i>m$ and $n\ge m$, then ${\mathcal{H}}_{{\mu }_n}{T}_{\sigma }{⨂}_{{\mathcal{H}}_{{\lambda }_n}}{W}_i=0$. So, T is surjective for all $n\ge m$. We have proved this lemma. □

From the above lemma we have the following corollary.

Corollary 1.

The surjective degree of a consistent sequence is less than its generated degree.

4. Weight

Let us recall some basic results about the modules of Hecke algebra [15]. Assume that q is not a root of unity; then, the Hecke algebra $\mathcal{H}$ is semisimple, and all the non-isomorphic simple $\mathcal{H}$-modules are indexed by all partitions $\lambda$. For each partition $\lambda$, the corresponding irreducible module is denoted by $S^{\lambda }$. By the Corollary 6.2 in Reference [15], the branching rule of Hecke algebra is the same as the classical branching rule for ${\mathfrak{S}}_n$-representations. The results about the branching rule for ${\mathfrak{S}}_n$-representations, stated in Reference [1], are also true for the modules of Hecke algebra. For $\lambda ⊢n$ and $m\in \mathbb{N}$, write $\lambda \to \mu$ if $\mu$ is a partition of $n+m$ such that $[\mu ]$ is obtained from $[\lambda ]$ by adding m boxes from different columns. We write that lemma of the branching rule in the following:

Lemma 4

(The branching rule [15]). Let λ be a partition of n and $m\in \mathbb{N}$. $Q_n$ is defined in the Stability degree Section 3 under Lemma 1. Then,

• ${Ind}_{{\mathcal{H}}_n⨂{\mathcal{H}}_m}^{{\mathcal{H}}_{n+m}}{S}^{\lambda }\cong \underset{\lambda \to \mu }{⨁}{S}^{\mu }$;
• $({Res}_{{\mathcal{H}}_{n-m}⨂{\mathcal{H}}_m}^{{\mathcal{H}}_n}{S}^{\lambda })/Q_m\cong \underset{\mu \to \lambda }{⨁}{S}^{\mu }$.

Lemma 5

([1]). Let λ be a partition of n and $a\le n$. Then,

• $S{(\lambda )}_n/Q_{n-a}=0⟺n-a>n-\left|\lambda \right|$.
• $S{(\lambda )}_n/Q_{n-\left|\lambda \right|}\cong S^{\lambda }$.
• If $n\ge a+\left|\lambda \right|$, then $S{(\lambda )}_n/Q_{n-a}$ is independent of n.
• Let V be a consistent sequence with weight less or equal to d. If $W_n/Q_{n-d}=0$ for any subquotient $W_n$ of $V_n$, then $W_n=0$.

Definition 7.

Let V be an $F{I}_{\mathcal{H}}$-module. We say a partition λ occurs in V, if there exists $n\ge 0$, $S{(\lambda )}_n$ occurs in the ${\mathcal{H}}_n$-module $V_n$. The weight of V is defined the maximum of $\left|\lambda \right|$ which λ occurs in V.

Lemma 6.

For any partition $\lambda ⊢m$, the $F{I}_{\mathcal{H}}$-module $M(S^{\lambda })$ has weight m.

Proof. 

By the definition of $M(S^{\lambda })$, $M{(S^{\lambda })}_n={\mathcal{H}}_n\underset{{\mathcal{H}}_{m,n-m}}{⨂}{S}^{\lambda }=Ind{S}^{\lambda }$. So, for any $n\ge m+{\lambda }_1$, adding $n-m$ boxes in different columns we obtain the partition $\lambda [n]$ occurs in $M(S^{\lambda })$. So, the weight of $M(S^{\lambda })$ is at least m. On the other hand, for any partition $\mu$ that occurs in $M{(S^{\lambda })}_n$ satisfies ${\mu }_1\ge n-m$. If we write $\mu =\nu [n]$ for some partition $\nu$, then we have $\left|\nu \right|=n-{\mu }_1\le m$. Then, the weight of $M(S^{\lambda })$ is at most m. We, thus, complete the proof of this lemma. □

If V is generated in degree $\le d$, then the weight of V is less or equal to d by the above lemma.

Lemma 7.

For any partition $\lambda =({\lambda }_1,{\lambda }_2,\cdots )$, the consistent sequence $M(S^{\lambda })$ has stability degree ${\lambda }_1$.

Proof. 

From our previous results, we know the stability degree of $M(S_{\lambda })$ is less than or equal to $\left|\lambda \right|$, and we have

$${\Phi }_{a}M{(S_{\lambda })}_n\cong \underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}(({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{S}^{\lambda })/Q_n),$$

(1)

where ${\mu }_n=(1,1,\cdots ,1,n)$ and ${\lambda }_n=(|\lambda |,a+n-\left|\lambda \right|)$.

By Lemma 2 and an easily computing, for $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$ there is some $k\le \left|\lambda \right|$ such that

$${\sigma }^{-1}{S}_{{\mu }_n}\sigma \cap S_{\{1,2,\dots ,|\lambda |\}}=S_{\{|\lambda |-k+1,\dots ,|\lambda |\}}.$$

For any $w\in S_{\{|\lambda |-k+1,\cdots ,|\lambda |\}}$, there exists $u\in S_{{\mu }_n}$ such that $w={\sigma }^{-1}u\sigma \in {\sigma }^{-1}{S}_{{\mu }_n}\sigma \cap S_{{\lambda }_n}$. Since $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}$, so $l(\sigma w)=l(\sigma )+l(w)$ and $l(u\sigma )=l(u)+l(\sigma )$. Assume x is an element in $S^{\lambda }$, and then we have

$$[T_{\sigma }\otimes T_{w}x]=[T_{\sigma }(T_{\sigma }^{-1}{T}_u{T}_{\sigma })\otimes x]=[T_u{T}_{\sigma }\otimes x]=[T_{\sigma }\otimes x].$$

By Lemma 5, for $k>{\lambda }_1$, we know $S^{\lambda }/Q_k=0$. So, the summands in the decomposition of (1) are stable when $n\ge {\lambda }_1$. Since $S^{\lambda }/Q_{{\lambda }_1}\ne 0$, then T is not surjective for $n={\lambda }_1-1$. We have completed the proof of this lemma. □

5. Noetherian Property

In this section, we will study the Noetherian property of $F{I}_{\mathcal{H}}$-modules. Assume that $F{I}_{\mathcal{H}}$-module V is finitely generated, and then the degree of V is also finite. Conversely, let V have degree $\le d$. If, in addition, $V_n$ for $n\le d$ is finitely generated ${\mathcal{H}}_n$-modules, then V is finitely generated.

Definition 8.

Given $a\ge 0$. Let $V=(V_n,{\varphi }_n)$ be an $F{I}_{\mathcal{H}}$-module. We define an $F{I}_{\mathcal{H}}$-module $S_{+a}V$ by ${(S_{+a}V)}_n=V_{n+a}$.

After a simple calculation, we derive that ${(S_{+a}V)}_n$ is isomorphic to ${Res}_{{\mathcal{H}}_n}^{{\mathcal{H}}_{n+a}}{V}_{n+a}$ as ${\mathcal{H}}_n$-modules.

Lemma 8.

Let $m\ge 1$ and $a\ge 0$. Then, there exists a decomposition of $S_{+a}M(m)$

$$S_{+a}M(m)=M(m)\oplus C_a,$$

where $C_a$ is finitely generated in degree $\le m-1$.

Proof. 

As we have computed in the previous sections

$$\begin{array}{cc}\hfill {(S_{+a}M(m))}_n& =M{(m)}_{n+a}\hfill \\ \hfill & ={\mathcal{H}}_{a+n}\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{\mathcal{H}}_m\hfill \\ \hfill & =\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{\mathcal{H}}_m),\hfill \end{array}$$

where ${\mu }_n=(1,1,\cdots ,1,n)$ and ${\lambda }_n=(m,a+n-m)$.

Let $\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}-\{(1)\}$. There are many choices for the representative elements of the double coset ${\mathcal{D}}_{{\mu }_n,{\lambda }_n}$. Since the number of elements of symmetric group ${\mathfrak{S}}_n$ is finite, we can always choose the representative elements with minimum length. There exists $m^{\prime }\in \{1,2,\cdots ,m-1\}$ such that ${\sigma }^{-1}{\mathfrak{S}}_{{\mu }_n}\sigma \cap {\mathfrak{S}}_m=S_{\{m^{\prime }+1,\cdots ,m\}}$. Then, we can decompose ${\mathfrak{S}}_m={⨆}_{{\sigma }^{\prime }\in {\mathcal{D}}^{\prime }}{S}_{\{m^{\prime }+1,\cdots ,m\}}{\sigma }^{\prime }{\mathfrak{S}}_{m^{\prime }}$, where ${\mathcal{D}}^{\prime }$ is the corresponding representative elements of minimal length. So, we have

$$\begin{array}{cc}\hfill \underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}}{⨁}({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{\mathcal{H}}_m)& =({\mathcal{H}}_{{\mu }_n}\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{\mathcal{H}}_m)⨁\left(\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}-\{(1)\}}{⨁}(\underset{{\sigma }^{\prime }\in {\mathcal{D}}^{\prime }}{⨁}({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }{T}_{{\sigma }^{\prime }}\underset{{\mathcal{H}}_{{\lambda }_n}}{⨂}{\mathcal{H}}_{m^{\prime }}))\right)\hfill \\ \hfill & \cong M{(m)}_n\oplus {(C_a)}_n,\hfill \end{array}$$

where we denote ${(C_a)}_n=\left(\underset{\sigma \in {\mathcal{D}}_{{\mu }_n,{\lambda }_n}-\{(1)\}}{⨁}(\underset{{\sigma }^{\prime }\in {\mathcal{D}}^{\prime }}{⨁}({\mathcal{H}}_{{\mu }_n}{T}_{\sigma }{T}_{{\sigma }^{\prime }}{⨂}_{{\mathcal{H}}_{{\lambda }_n}}{\mathcal{H}}_{m^{\prime }}))\right)$.

Now, it is not difficult to obtain that $C_a$ is finitely generated in degree $\le m-1$ and we complete the proof of this lemma. □

From the above lemma, we know the degree of $S_{+a}M(m)$ is finite and by Lemma 1 $M(m)$ can be replaced by a consistent sequence with finite degree. Conversely, according to our definition of $S_{+a}V$, we have:

Lemma 9.

Let V be a consistent sequence. If $S_{+a}V$ is of finite degree, then so is V.

Theorem 1 (Noetherian Property).

Let $V=(V_n,{\varphi }_n)$ be finitely generated and $W\subseteq V$. Then, W is also finitely generated.

Proof. 

Suppose V is generated in degree m. By Lemma 1, V is a quotient of a finite direct sum of $M(m_i)$ for $m_i\le m$. So, we need to prove $M(m_i)$ satisfies the Noetherian property. Let $W\subseteq M(m)$. Since the injective degree of $M(m)$ is zero, we can assume that $M{(m)}_n\subseteq M{(m)}_{n+1}$ for all n. Since every $W_n$ is finite dimensional, then W has finite degree implies W is finitely generated. To show $deg(W)<\infty$, it suffices to prove it for $S_{+a}W$ for some a.

By Lemma 8, $S_{+a}M(m)$ can be decomposed as two summand, one of degree m. In addition, so is $S_{+a}W$. Since the map ${\varphi |}_{W_n}:W_n\to W_{n+1}$ is injective, we can assume $W_n\subseteq W_{n+1}$. For any a, ${\mathcal{H}}_m=M{(m)}_m$ is contained in $S_{+a}M{(m)}_m$, which is exactly the summand of degree m. Let $W_m^a$ be the summand of $S_{+a}{W}_m$. Then, $W_m^a\subseteq W_m^{a+1}\subseteq {\mathcal{H}}_m$. Since $\mathcal{H}$ is finite dimensional, there exists N such that $W_m^N=W_m^{N+1}$. For any $a\ge 0$, $W_m^{N+a}$ generates $W_{m+a}^N$ which implies that $W^N$ is finitely generated. For the degree $\le m-1$ of $S_{+N}W$, we use induction, which implies $S_{+N}W$ is finitely generated of degree. □

6. Representation Stability

The following lemma shows that the representation stable range $N<\infty$ if the stability degree and weight are finite.

Lemma 10.

Let $V=(V_n,{\varphi }_n)$ be an $F{I}_{\mathcal{H}}$-module with stability degree s and weight m. Then, V is uniformly representation stable with stable range $n\ge s+m$.

Proof. 

Let ${\varphi }_n^{\prime }:{\mathcal{H}}_{n+1}{⨂}_{{\mathcal{H}}_n}{V}_n\to V_{n+1}$.

In order to indicate the injective and surjectivity, that is to say, $ker{\varphi }_n=0$ and $coker{\varphi }_n^{\prime }=0$ for $n\ge s+m$, it suffices to prove $ker{\varphi }_n/Q_{n-m}=0$ and $coker{\varphi }_n^{\prime }/Q_{n+1-m}$$=0$ by Lemma 5.

Since $n-m\ge s$, the map $T:{\Phi }_m{(V)}_{n-m}\to {\Phi }_m{(V)}_{n+1-m}$ is bijective. Moreover, notice that the diagrams

and

commute. So, $T_1$ is injective and $T_2^{\prime }$ is surjective. We have obtained that $ker{\varphi }_n/Q_{n-m}=0$ and $coker{\varphi }_n^{\prime }/Q_{n+1-m}=0$.

Assume that $V_n={⨁}_{\lambda }{c}_{\lambda ,n}S{(\lambda )}_n$, and we need to show $c_{\lambda ,n}$ is independent of n when $n\ge s+m$. Since the weight of V is m, there is no $\lambda$ with $\left|\lambda \right|>m$ occurred in V. By Lemma 5, $S{(\lambda )}_n/Q_{n-m}=0$ when $\left|\lambda \right|>m$. So,

$$\begin{array}{cc}\hfill V_n/Q_{n-m}& =\underset{\left|\lambda \right|\le m}{⨁}{c}_{\lambda ,n}(S{(\lambda )}_n/Q_{n-m})\hfill \\ \hfill & =\underset{\left|\lambda \right|<m}{⨁}{c}_{\lambda ,n}(S{(\lambda )}_n/Q_{n-m})⨁(\underset{\left|\lambda \right|=m}{⨁}{c}_{\lambda ,n}(S{(\lambda )}_n/Q_{n-m}))\hfill \\ \hfill & =\underset{\left|\lambda \right|<m}{⨁}{c}_{\lambda ,n}(S{(\lambda )}_n/Q_{n-m})⨁(\underset{\left|\lambda \right|=m}{⨁}{c}_{\lambda ,n}{S}^{\lambda }).\hfill \end{array}$$

Because $n-m$ is greater or equal to the stability degree and for the first term of the above decomposition by the induction, we obtain $c_{\lambda ,n}$ is independent of n. □

Now, we state and prove our main theorem about representation stability of the representations of Hecke algebras.

Theorem 2.

An $F{I}_{\mathcal{H}}$-module $V=(V_n,{\varphi }_n)$ is finitely generated if and only if the sequence $(V_n,{\varphi }_n)$ is uniformly representation stable and each $V_n$ is finite-dimensional.

Proof. 

Assume V is uniformly stable with range N. Because of the surjectivity of V, we have $V_n=span{(im{\varphi }_{n-1})}_n$ for all $n-1\ge N$. So, $span({⨆}_{n=0}^N{V}_n)=V$ and, together with the finite-dimensional, implies that V is finitely generated.

For the converse, by Lemma 10, we only need to show V has finite stability degree and weight. According to Corollary 1, the surjective degree is finite. Assume that V is generated in degree $\le d$. There exists an epimorphism $g:M\to V$, where $M=M(W)$ for some $W={(W_n)}_{n\ge 0}$. Let K be the kernel of g. By Theorem 1 and Corollary 1, K is also finitely generated, and the surjective degree of K is finite, say s.

For given $a\ge 0$ and for any $n\ge s$, we have the following commutative diagram:

$$\begin{array}{ccccccccc}0& \to & {\Phi }_a{(K)}_n& \to & {\Phi }_a{(M)}_n& \to & {\Phi }_a{(V)}_n& \to & 0\\ & & T_2\downarrow \phantom{T_2}& & T_1\downarrow \phantom{T_1}& & T\downarrow \phantom{T}& & \\ 0& \to & {\Phi }_a{(K)}_{n+1}& \to & {\Phi }_a{(M)}_{n+1}& \to & {\Phi }_a{(V)}_{n+1}& \to & 0.\end{array}$$

By the Snake Lemma, we have an exact sequence:

$$\ker T_1\to \ker T\to \mathrm{coker}{T}_2\to \mathrm{coker}{T}_1\to \mathrm{coker}T\to 0.$$

So, $ker{T}_1=0$ and $coker{T}_2=0$ imply $kerT=0$. This shows the injective degree of V is finite. We, thus, have completed the proof. □

Remark 2.

As an application of Theorem 2, we point out that the result for representation stability of symmetric group representations of Theorem 1.13 in Reference [1] can be derived by our theorem as q tends to 1.

7. Conclusions

We study the free object, projective object, surjective degree, injective degree, and stability degree of the abelian category $F{I}_{\mathcal{H}}$-modules. We prove that $F{I}_{\mathcal{H}}$-modules satisfies the Noetherian property. In addition, we obtain that an $F{I}_{\mathcal{H}}$-module $V=(V_n,{\varphi }_n)$ is finitely generated if and only if the sequence $(V_n,{\varphi }_n)$ is uniformly representation stable and each $V_n$ is finite-dimensional. As q tends to 1, our results can derive the stability for representations of symmetric groups. As for the subsequent work of $F{I}_{\mathcal{H}}$-modules, one can further study homological stability of the representation category of Hecke algebras. Moreover, the stabilities for representations of quantum groups and Hecke algebras of other types are also worth studying. In addition, then, one can investigate the relationship with Schur-Weyl duality.