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
-modules, where
is the category of finite sets and injective maps. Essentially,
-module over a commutative ring
R is a functor from the category
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
-module. Many important results have been obtained, such as homological properties of
-modules, Noetherian properties, and representation stability properties, in a series of articles [
3,
4,
5,
6,
7,
8].
Let
be one of the symmetric group
, the general linear group
, or the special linear group
. In 2013, Church and Farb introduced the theory of representation stability to study the representations
of these natural groups
[
9]. Let us review some basic concepts before introducing relevant background. The idea is mainly to study the consistent sequence of representations
of groups
as follows. A sequence of
-representations,
is called consistent sequence if, for all
and
, the diagram
is commutative, where each
is a linear map.
A consistent sequence is called representation stable if, for sufficiently large n, it satisfies
Church, Ellenberg, and Farb used the theory of
-modules to study the consistent sequence of representations of the symmetric groups
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
-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
-modules over Noetherian rings. The consistent sequence of representations of symmetric groups given by a finitely generated
-module turned out to be representation stable.
In Reference [
10], Gan and Watterlond researched
-modules, where
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
, where
q is a power of prime. That is, a
-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
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
for the 0-Hecke algebra
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
-module.
Hecke algebra of type A is the deformation of group algebra , and the representations of is also the deformation of representations of . In the light of works of Church, Ellenberg, and Farb, the relation between the representation categories of and strongly suggests the existence of representation stability of Hecke algebras . 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 -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. is not a root of unity. Let and be the symmetric group. It is easy to show that is generated by the transpositions . For the convenience of writing, we denote the transposition by for . The Iwahori-Hecke algebra (or simply the Hecke algebra) of the symmetric group is the unital associative k-algebra with generators and relations
, for ;
, for ;
, for .
We shall denote .
Suppose
. Then,
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
is denoted by
; thus, we have
Denote which is an element of the Hecke algebra . Then, is free with k-basis . If and , then it is not difficult to obtain that .
We now review the concept of composition (partition) of natural numbers, which will be used in later sections. Let
. A sequence of non-negative integers
which satisfies
is called a composition of
n. A composition
is called a partition of
n when
. Composition (partition)
of
n is generally denoted by
(
). Let
and
. Obviously, the set
is a subgroup of
. For each composition
of
n, the standard Young subgroup
of
is defined by the direct product
. For example, if
is a composition of 6, then the corresponding Young subgroup is
. For more details of Young subgroups, please refer to Reference [
13]. The subalgebra
of
generated by
is
k-free with basis
. The subalgebras of these types are the so-called Young subalgebras.
Let
be a partition of some natural number. For any
, the padded partition is defined as
where
.
In the case of the Hecke algebra 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 and , we denote the corresponding simple -module and set .
Since is naturally regarded as a subgroup of , then the generators of Hecke algebra is a subset of generators of Hecke algebra . That is to say, is a subalgebra of . The natural injective k-algebra map from to is denoted by .
Definition 1. A sequence is called consistent sequence or -module if it satisfies: for all , the diagramis commutative, where is -module, and is k-linear map. Let
and
be two
-modules. A morphism
from
V to
W is a sequence of homomorphisms
for all
such that the following diagram
is commutative.
Remark 1. -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 be a morphism. We need to find an -module, and the kernel of this morphism satisfies the kernel’s universal property of this category. As usual, this -module is denoted by . In addition, the construction of the -module is through with defining at each point n. That is, is defined by , and the map from to is the restriction of . Now, it is easy to check that, for all , we have commutative composite mappings, i.e., , which makes an -module. Similarly, we can also derive the tensor product of two -modules straightforward. The tensor product of two -modules and is defined by , and the map from to is defined by .
Definition 2. Let , where is an -module. Denote the composition for . Then, will be an -module under the action , where for and and . We define a sequence for ,with the linear map determined by , where and . It is easy to check that is an -module. If satisfies for some m and for , then we denote by .
Let be an -module. Denote for and . For a disjoint union, , where . Next, we define the concept of generation through the span of linear space. We define to be the submodule of generated by . In addition, the map is the restriction of . It is trivial to check from the above discussion that is an -module.
Definition 3. We say that an -module V is generated in degree if V is generated by elements of for .
According to the above definition, if
V is generated in degree
, then we have
Definition 4. We say that an -module V is finitely generated if there is a finite set S of the disjoint union such that .
Definition 5 (Representation stability of consistent sequence)
. Let be an -module. This sequence is called uniformly representation stable if there exists an integer N such that for each with , the following conditions hold
Injectivity: is injective;
Surjectivity: is generated by the image of ;
Multiplicities: , where the multiplicities 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
-representations is representation stable can be converted to a finite generation property for an
-module.
3. Stability Degree
In this section, we define the concepts of stability degree, injective degree and surjective degree for an -module. For , we will compute these numbers for the -module . The following lemma shows that is similar to the “free object”.
Lemma 1. If an -module is generated in degree , then there exists an epimorphism from to V.
Proof. For every , is a summand of since by definition. So, the functions from to are defined by projections. Now, let , and then . If for , then . The fact implies that is a morphism. From the construction of f and V generated in degree , we know f is an epimorphism. □
Given and an -module , we define a sequence as follows. Denote by the subspace of spanned by for and then define . In addition, the map is defined by . The map T is well-defined since, for , .
Definition 6. If there exists a natural number s such that for all , the map defined above is an isomorphism for , 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(or inj-deg, sur-deg).
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 . For and , let and be two compositions of . Denote the representative elements of minimal length of the double cosets . Then, and if , the set is independent of n.
Proof. By Lemma 1.7 in Reference [
14], there is a bijection from the set of row-standard tableaux in
to
. These row-stantard
-tableaux of type
are determined by the subset of
which is in the first row of
. Denote the number of the set of row-standard tableaux in
by
. So, we have
. If
is the element of minimal length in a double coset
, then
’s length is also minimal in the double coset
. We obtain that
implies
. If
, the number
is independent of
n. So, we complete the proof of the lemma. □
For
, there exist
,
, and
such that
. According to the Lemma 1.6 in Reference [
14], these elements can be chosen to satisfy
. This property will be used in the following lemma.
Lemma 3. Assume , where is an -module for . Then, has injective degree 0; if for all , then we have surj-deg. Therefore, has injective degree 0 and surjective degree m.
Proof. At first, we compute
by definition.
The map
is the direct sum of the maps: For
,
defined by
, where
and
. In addition, so
The map restricted in one factor in the above decomposition is the identity. So, we have obtained that the inj-deg. If for and , then . So, T is surjective for all . 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
is semisimple, and all the non-isomorphic simple
-modules are indexed by all partitions
. For each partition
, the corresponding irreducible module is denoted by
. By the Corollary 6.2 in Reference [
15], the branching rule of Hecke algebra is the same as the classical branching rule for
-representations. The results about the branching rule for
-representations, stated in Reference [
1], are also true for the modules of Hecke algebra. For
and
, write
if
is a partition of
such that
is obtained from
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 . is defined in the Stability degree Section 3 under Lemma 1. Then, Lemma 5 ([
1])
. Let λ be a partition of n and . Then,.
.
If , then is independent of n.
Let V be a consistent sequence with weight less or equal to d. If for any subquotient of , then .
Definition 7. Let V be an -module. We say a partition λ occurs in V, if there exists , occurs in the -module . The weight of V is defined the maximum of which λ occurs in V.
Lemma 6. For any partition , the -module has weight m.
Proof. By the definition of , . So, for any , adding boxes in different columns we obtain the partition occurs in . So, the weight of is at least m. On the other hand, for any partition that occurs in satisfies . If we write for some partition , then we have . Then, the weight of is at most m. We, thus, complete the proof of this lemma. □
If V is generated in degree , then the weight of V is less or equal to d by the above lemma.
Lemma 7. For any partition , the consistent sequence has stability degree .
Proof. From our previous results, we know the stability degree of
is less than or equal to
, and we have
where
and
.
By Lemma 2 and an easily computing, for
there is some
such that
For any
, there exists
such that
. Since
, so
and
. Assume
x is an element in
, and then we have
By Lemma 5, for
, we know
. So, the summands in the decomposition of (
1) are stable when
. Since
, then
T is not surjective for
. We have completed the proof of this lemma. □
5. Noetherian Property
In this section, we will study the Noetherian property of -modules. Assume that -module V is finitely generated, and then the degree of V is also finite. Conversely, let V have degree . If, in addition, for is finitely generated -modules, then V is finitely generated.
Definition 8. Given . Let be an -module. We define an -module by .
After a simple calculation, we derive that is isomorphic to as -modules.
Lemma 8. Let and . Then, there exists a decomposition of where is finitely generated in degree . Proof. As we have computed in the previous sections
where
and
.
Let
. There are many choices for the representative elements of the double coset
. Since the number of elements of symmetric group
is finite, we can always choose the representative elements with minimum length. There exists
such that
. Then, we can decompose
, where
is the corresponding representative elements of minimal length. So, we have
where we denote
.
Now, it is not difficult to obtain that is finitely generated in degree and we complete the proof of this lemma. □
From the above lemma, we know the degree of is finite and by Lemma 1 can be replaced by a consistent sequence with finite degree. Conversely, according to our definition of , we have:
Lemma 9. Let V be a consistent sequence. If is of finite degree, then so is V.
Theorem 1 (Noetherian Property)
. Let be finitely generated and . 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 for . So, we need to prove satisfies the Noetherian property. Let . Since the injective degree of is zero, we can assume that for all n. Since every is finite dimensional, then W has finite degree implies W is finitely generated. To show , it suffices to prove it for for some a.
By Lemma 8, can be decomposed as two summand, one of degree m. In addition, so is . Since the map is injective, we can assume . For any a, is contained in , which is exactly the summand of degree m. Let be the summand of . Then, . Since is finite dimensional, there exists N such that . For any , generates which implies that is finitely generated. For the degree of , we use induction, which implies is finitely generated of degree. □
6. Representation Stability
The following lemma shows that the representation stable range if the stability degree and weight are finite.
Lemma 10. Let be an -module with stability degree s and weight m. Then, V is uniformly representation stable with stable range .
Proof. Let .
In order to indicate the injective and surjectivity, that is to say, and for , it suffices to prove and by Lemma 5.
Since
, the map
is bijective. Moreover, notice that the diagrams
and
commute. So,
is injective and
is surjective. We have obtained that
and
.
Assume that
, and we need to show
is independent of
n when
. Since the weight of
V is
m, there is no
with
occurred in
V. By Lemma 5,
when
. So,
Because is greater or equal to the stability degree and for the first term of the above decomposition by the induction, we obtain is independent of n. □
Now, we state and prove our main theorem about representation stability of the representations of Hecke algebras.
Theorem 2. An -module is finitely generated if and only if the sequence is uniformly representation stable and each is finite-dimensional.
Proof. Assume V is uniformly stable with range N. Because of the surjectivity of V, we have for all . So, 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 . There exists an epimorphism , where for some . 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
and for any
, we have the following commutative diagram:
By the Snake Lemma, we have an exact sequence:
So, and imply . 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.