1 Introduction

Homological stability is a topological property of certain sequences of topological spaces: given a sequence \(\{X_n\}_{n=1}^\infty \) of topological spaces with maps \(\phi _n : X_{n} \rightarrow X_{n+1}\) for each n then \(\{X_n\}_{n=1}^\infty \) is homologically stable if there exists a positive N so that \((\phi _n)_*: H_i(X_{n}) \rightarrow H_i(X_{n+1})\) is an isomorphism whenever \(n \ge N\). In other words the homology groups stabilize after a certain point in this sequence; however, the topological spaces change later in the sequence, the changes do not affect the ith homology group.

Church and Farb [9], and later Church, Ellenberg, and Farb [6] defined representation stability to mimic the topological definition. Informally, the sequence of \(S_n\)-representations \(V_1 {\mathop {\rightarrow }\limits ^{f_1}} V_2 {\mathop {\rightarrow }\limits ^{f_2}} V_3 {\mathop {\rightarrow }\limits ^{f_3}} \cdots \) is representation stable if there are \(S_n\)-equivariant linear injections \(f_n: V_n \rightarrow V_{n+1}\) and if there exists an N so that the multiplicities \(c_{\mu ,n}\) in the decomposition into irreducibles

$$\begin{aligned} V_n = \bigoplus _\mu c_{\mu ,n} V(\mu )_n \end{aligned}$$

are independent of n for all \(n \ge N\). (Section. 6 defines representation stability precisely.)

One important problem is to find families of representations that are representation stable; these families often arise from geometric considerations. In their original work, Church, Ellenberg, and Farb identify a number of representation stable families arising from geometry/topology and classical representation theory [6]. Indeed their example of the diagonal coinvariant algebra [6, Sect. 5] is the sequence of cohomology rings of flag varieties \(\{H^*(GL_n/B_n)\}_{n=1}^\infty \) which in some sense is the springboard of this paper. Since then, others have demonstrated that representation stability arises naturally in many contexts, including arrangements associated to root systems [2], linear subspace arrangements [15], configuration spaces in \(\mathbb {R}^d\) [17], filtrations of Torelli groups [24], moduli spaces of Riemann surfaces of genus g with n labeled marked points [26], and others [11, 27].

The central goal of this paper is to prove that an important family of \(S_n\)-representations called Springer representations is representation stable. The Springer representation is the archetypal geometric representation: In its most basic form, it arises when the symmetric group \(S_n\) acts on the cohomology of a family of subvarieties of the flag variety called Springer fibers. The flag variety can be described as the set of nested vector subspaces

$$\begin{aligned} V_1 \subseteq V_2 \subseteq \cdots \subseteq V_{n-1} \subseteq \mathbb {C}^n \end{aligned}$$

where each \(V_i\) is i-dimensional. Given a nilpotent matrix X, the Springer fiber consists of the flags that are fixed by X in the sense that \(XV_i \subseteq V_i\) for all i. Every nilpotent matrix is conjugate to one in Jordan form, which is determined by a partition of n into Jordan blocks. Since Springer fibers associated to conjugate matrices X and \(gXg^{-1}\) are homeomorphic, the Springer fibers are parametrized by partitions of n.

Springer first constructed a representation of the symmetric group \(S_n\) on the cohomology of Springer fibers [28]. Since then, the representation has been recreated in many different ways [4, 10, 22, 29]. The geometry of Springer fibers encodes key data about representations of the symmetric group. For instance, the top-dimensional cohomology is an irreducible \(S_n\)-representation [28], and the ungraded representation on \(H^*(Spr_{\lambda })\) is Young’s representation associated to the partition \(\lambda \) [16, Introduction].

Our analysis of Springer representations uses the co-FI category, which Church, Ellenberg, and Farb defined to concisely describe the compatibility conditions needed for representation stability. Theorem 3 proves that for any increasing sequence of Young diagrams, the corresponding Springer representations form a graded co-FI-module of finite type. The main consequence from our point of view is that sequences of Springer representations are representation stable (see Corollary 3 for a precise statement).

From this many other properties follow: (1) for each fixed degree and n large enough, the character is given by a polynomial that is independent of n, and in particular (2) the dimension of the Springer representation is eventually polynomial in n. It is this second consequence that we explore in Sect. 7.

Remark 1

Curiously, the literature does not make a clear distinction between the Springer representation and its dual. Hotta first observed this and classified existing constructions of the Springer representation up to that point [18]. However, the ambiguity persisted with subsequent constructions of the Springer representation. In this paper, we treat “the” Springer representation interchangeably with its dual. We prove that the Garsia–Procesi construction of the Springer representation is a co-FI-module (graded, of finite type) and its dual is representation stable. Properties like dimension and decomposition into irreducibles are well-behaved with respect to duality and apply to the original Garsia–Procesi construction, too.

Kim proves a different kind of stability of Springer representations, giving conditions under which the lower-graded parts of the Springer representation for \(\lambda \) coincide with those of \(\lambda '\) for partitions \(\lambda , \lambda '\) of the same n [20]. Kim recovers Theorem 6 using his notion of stability [20, Corollary 4.3].

This paper is organized as follows. In Sect. 2, we define the conditions that guarantee the stability of a sequence of \(S_n\)-representations. In Sect. 3, we describe Garsia–Procesi’s combinatorial description of the Springer representation, and in particular the so-called Tanisaki ideal. In Sect. 4, we prove there is no FI-module structure on sequences of Springer representations, except for the trivial representation. On the other hand, in Sect. 5 we prove that the sequence of Tanisaki ideals forms a co-FI-ideal and deduce that a co-FI-module structure exists on all sequences of Springer representations. In Sect. 6, we conclude with our main result on the stability of the Springer representations. We give some concrete combinatorial consequences of this stability in Sect. 7, and a collection of open questions in Sect. 8 that probe the new combinatorial ideas raised by representation stability.

2 FI and co-FI

In this section, we describe FI-modules and co-FI-modules, defined by Church, Ellenberg, and Farb [6] to streamline and extend the essential features of Church and Farb’s earlier notion of representation stability [9]. We begin with the categories FI and co-FI, which carry actions of the permutation groups \(S_n\) and are constructed to be compatible with inclusions. The key example of FI- and co-FI-modules for this manuscript is the sequence of polynomial rings \(\{k[x_1,\ldots ,x_n]\}\). We then list the properties about FI- and co-FI-modules that we will need to establish that sequences of Springer representations are graded co-FI-modules.

This section provides only what is needed in this paper. The interested reader is referred to Church, Ellenberg, and Farb’s work for many other interesting results [6].

Remark 2

We assume k is a field of characteristic zero. Parts of Church-Ellenberg-Farb’s theory extend to other fields as well [6].

Definition 1

(FI-module, FI-algebra graded FI-algebra) Let \(\mathbf {n}\) denote the set \(\{1,\ldots ,n\}\). FI is the category whose objects are finite sets and whose morphisms are injections. This is equivalent to the category whose objects are sets \(\mathbf {n} = \{1,2,\ldots ,n\}\) and whose morphisms are injections \(\mathbf {m} \rightarrow \mathbf {n}\). An FI-module (FI-algebra, graded FI-algebra) over a commutative ring k is a functor V from FI to the category of modules over k (k-algebras, graded k-algebras). We usually denote the k-module (respectively, algebra) \(V(\mathbf {n})\) by \(V_n\).

Modules that carry permutation actions compatible with the permutation action on integers provide a rich source of examples of FI-modules. The next example is the most important for our purposes.

Example 1

([6, Example 4.1.2 and Remark 4.1.3]) Define a functor R by:

  • R sends the object \(\mathbf {n}\) to the polynomial ring \(k[x_1,\ldots ,x_n]\) and

  • R sends the morphism \(f: \mathbf {m} \rightarrow \mathbf {n}\) to the homomorphism

    $$\begin{aligned} f_*: k[x_1,\ldots ,x_m] \rightarrow k[x_1,\ldots ,x_n] \end{aligned}$$

    induced by the condition that \(f_*(x_i)=x_{f(i)}\) for all \(i \in \mathbf {m}\).

Then, R is a graded FI-algebra. In particular, we have that \(R_n\) is the polynomial ring \(k[x_1,\ldots ,x_n]\).

The category co-FI is opposite to FI. The polynomial algebras also form a graded co-FI-algebra, as described below.

Definition 2

(co-FI-module, co-FI-algebra, graded co-FI-algebra) Denote the opposite category of FI by co-FI. In particular, the objects in co-FI are finite sets, without loss of generality the sets \(\mathbf {n}\), and the morphisms in co-FI from \(\mathbf {n}\) to \(\mathbf {m}\) are the morphisms in FI from \(\mathbf {m}\) to \(\mathbf {n}\). A co-FI-module (co-FI-algebra, graded co-FI-algebra) over a commutative ring k is a functor from co-FI to the category of k-modules (k-algebras, graded k-algebras).

For instance, if V is an FI-module over k, then the dual \(V^\vee \) forms a co-FI-module, and vice versa.

Example 2

([6, Example 4.1.2 and Remark 4.1.3]) Define a functor R as follows:

  • R sends the object \(\mathbf {n}\) to the polynomial ring \(k[x_1,\ldots ,x_n]\) and

  • R sends the FI-morphism \(f: \mathbf {m} \rightarrow \mathbf {n}\) to the homomorphism

    $$\begin{aligned} f^*: k[x_1,\ldots ,x_n] \rightarrow k[x_1,\ldots ,x_m] \end{aligned}$$

    induced by the condition that

    $$\begin{aligned} f^*(x_i)=\left\{ \begin{array}{ll} x_{f^{-1}(i)} &{}\text { if } i \in \text {Im}(f) \text { and} \\ 0 &{} \text { otherwise.} \end{array}\right. \end{aligned}$$

Then, R is a graded co-FI-algebra.

The module categories FI and co-FI are abelian and so admit many of the algebraic constructions that modules and algebras do. In particular, we can consider co-FI-quotients of a co-FI-module and FI-submodules of an FI-module.

Example 3

([6, Classical coinvariant algebra in Sect. 5]) For each n let \(I_n\) denote the ideal of symmetric polynomials with no constant term. The sequence of ideals I is not an FI-submodule of the FI-module R because the image of an \(S_m\)-symmetric polynomial under the inclusion map \(\iota : \{1,\ldots ,m\} \rightarrow \{1,\ldots ,n\}\) is not symmetric under the larger group \(S_n\). However, the ideals I do form a co-FI-submodule of the co-FI-module R.

The next lemma proves that the image of a set of \(S_m\)-invariant polynomials under an arbitrary injection is the same as the image under the inclusion \(\iota \) that is the identity on the integers \(\{1,\ldots ,m\}\). We use it to simplify later calculations.

Lemma 1

Suppose that \(m \le n\) and that \(f: \{1,\ldots ,m\} \rightarrow \{1,\ldots ,n\}\) is an injection. Let \(\iota \) be the inclusion \(\iota : \{1,\ldots ,m\} \rightarrow \{1,\ldots ,n\}\) that sends \(\iota (i) = i\) for each \(1 \le i \le m\).

Consider the action of \(S_n\) on \(R_n\) under which for each \(w \in S_n\) and \(x_i \in R_n\) we have \(w(x_i) = x_{w(i)}\). Let \({\mathcal {J}} \subseteq R_n\) be any set of polynomials that are preserved under the \(S_{n}\)-action, in the sense that \(w {\mathcal {J}} \subseteq {\mathcal {J}}\) for all permutations \(w \in S_{n}\). Then,

$$\begin{aligned} f^*(\mathcal {J}) = \iota ^*(\mathcal {J}). \end{aligned}$$

Proof

Our hypothesis means that the polynomial \(p \in {\mathcal {J}}\) if and only if the image \(w(p) \in {\mathcal {J}}\) for each \(w \in S_{n}\). The map \(f: \{1,\ldots ,m \} \hookrightarrow \{1,\ldots ,n\}\) is an injection so we can define a permutation \(w_f \in S_{n}\) by

$$\begin{aligned} w_f = \left\{ \begin{array}{ll} f(i) \mapsto i &{} \text { for all }i\text { with } 1 \le i \le m \\ j_i \mapsto m+i &{} \text { for }j_i\text { such that both } j_1<j_2< \cdots <j_{n-m} \text { and } \\ &{} \quad \{ j_1, j_2, \cdots ,j_{n-m}\} \cup \text {Im}(f) = \{1,2,\ldots ,n\}. \end{array} \right. \end{aligned}$$

We know that for each i

$$\begin{aligned} \iota ^* (w_fx_i) = \iota ^* (x_{w_f(i)}) \end{aligned}$$

by definition of the \(S_n\)-action on permutations. By construction of \(\iota ^*\), we have

$$\begin{aligned} \iota ^* (x_{w_f(i)}) = \left\{ \begin{array}{ll} 0 &{} \text { if } i \in \{ j_1, j_2, \cdots ,j_{n-m}\} \\ x_{f^{-1}(i)} &{} \text { otherwise}. \end{array} \right. \end{aligned}$$

This is exactly \(f^*(x_i)\). Thus, \(f^*(p) = \iota ^* (w_fp)\) for all polynomials \(p \in {\mathcal {J}}\) and hence as desired \(f^*({\mathcal {J}}) = \iota ^*({\mathcal {J}})\). \(\square \)

It follows that if I is a sequence of symmetric homogeneous ideals in the graded co-FI-algebra R, then we can prove I is a co-FI-submodule simply by considering inclusions—or even just a subset of inclusions.

Corollary 1

Let R be a graded co-FI-algebra and I be a sequence of ideals \(\{I_1,I_2,\ldots \}\) with each \(I_n\) an \(S_n\)-invariant homogeneous ideal in \(R_n\). The following are equivalent:

  1. 1.

    I is a co-FI-submodule of R.

  2. 2.

    For each \(m<n\) and inclusion \(\iota : \{1,\ldots ,m\} \rightarrow \{1,\ldots ,n\}\) defined by \(\iota (i)=i\) for all i, the induced map satisfies \(\iota ^*(I_n)\subseteq I_m\).

  3. 3.

    For each n and inclusion \(\iota _n: \{1,\ldots ,n-1\} \rightarrow \{1,\ldots ,n\}\), the induced map satisfies \(\iota _n^*(I_n)\subseteq I_{n-1}\).

Proof

Part (1) is equivalent to Part (2) by Lemma 1. Part (3) implies Part (2) by definition. The composition of inclusions \(\iota _n \circ \iota _{n-1} \circ \cdots \circ \iota _{m+1}\) is the inclusion \(\iota : \{1,\ldots ,m\} \hookrightarrow \{1,\ldots ,n\}\). By functoriality if \(\iota _{i}^*(I_i) \subseteq I_{i-1}\) for each \(i\ge 2\), then \(\iota ^*(I_n) \subseteq I_m\) for each pair \(n \ge m \ge 1\). So Part (3) implies Part (2). \(\square \)

The definition of finitely generated FI-modules is crucial to representation stability. It differs importantly from the corresponding definition for modules or rings because it incorporates the underlying \(S_n\)-action. As we see in Example 4, this implies that FI-modules often have fewer generators than we might expect.

In fact, though this does not appear explicitly in the literature, the category of finitely generated FI-modules over Noetherian rings is also abelian (by the Noetherian property, proven over \(\mathbb {C}\) in [6] and over other Noetherian rings in [8]) [5]. This is the thrust of the arguments that we cite in this paper.

Definition 3

(Finite generation, finite type) An FI-module V is finitely generated if there is a finite set S of elements in \(\coprod _i V_i\) so that no proper sub-FI-module of V contains S. A graded FI-module V has finite type if the ith graded part \(V^i\) is finitely generated for each i. A graded co-FI-module W is of finite type if its dual \(W^\vee \) is a graded FI-module of finite type [6, Co-FI-algebras in Sect. 4.2]

Example 4

([6, Graded FI-modules of finite type in Sect. 4.2], [12, Example 4.2], [30, Example 1.4]) The sequence \(R = \{k[x_1,\ldots ,x_n]\}\) of polynomial rings are not finitely generated as an FI-module; indeed no ring \(k[x_1,\ldots ,x_n]\) is a finite dimensional k-vector space. However, when graded by polynomial degree, each graded part of the sequence R is finitely generated as an FI-module. For instance, the graded part of degree 3 is generated by \(x_1^3\), \(x_1^2x_2\), and \(x_1x_2x_3\) since every monomial of degree three is obtained by permuting indices of one of these three. More generally the graded part of degree d is generated by all monomials of the form \(x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n}\) over partitions \(d_1 \ge d_2 \ge \cdots \ge d_n \ge 0\) of d.

For completeness, we define the quotient of an FI-algebra or co-FI-algebra.

Definition 4

([7, Definition 2.76]) Let R be a graded FI-algebra. If I is a graded FI-submodule (respectively, co-FI-submodule) for which each object \(I_n\) is a homogeneous ideal in \(R_n\) then I is called an FI-ideal (respectively, co-FI-ideal). The quotient FI-module R/I is defined so that \((R/I)_n=R_n/I_n\) for each n (respectively, co-FI).

The following proposition is our main tool, and sketches the main points of [8, Theorem F]. This result implies that we only need to prove each sequence of Tanisaki ideals forms a co-FI-submodule; after that, a straightforward algebraic argument allow us to conclude that sequences of Springer representations are representation stable.

Proposition 1

Let R be a graded co-FI-module over a Noetherian ring and let I be a co-FI-ideal. Then, the dual \(\left( R/I\right) ^\vee \) is a graded FI-module. If \(R^\vee \) has finite type, then so does \(\left( R/I\right) ^\vee \).

Proof

The dual of a graded co-FI-module is a graded FI-module by purely formal properties. Suppose further that \(R^\vee \) has finite type and consider the jth graded parts \(\left( R_n\right) _j^\vee \) and \(\left( R_n/I_n\right) _j\) for each n. The dual \(\left( R_n/I_n\right) _j^\vee \) is the FI-submodule of \(\left( R_n\right) _j^\vee \) consisting of those functionals that vanish on the jth graded parts \(\left( I_n\right) _j = I_n \cap \left( R_n\right) _j\) by definition of graded quotients. Each FI-submodule of an FI-module of finite type over a Noetherian ring is also of finite type [8, Theorem A], thus proving the claim. \(\square \)

In Sects. 7 and 8 of this manuscript, we analyze properties of FI-modules in the case of Springer representations.

3 Springer theory

This section summarizes two key combinatorial tools: Biagioli–Faridi–Rosas’s description of generators for each Tanisaki ideal and Garsia–Procesi’s description of a basis for the Springer representation.

3.1 The Tanisaki ideal and its generators

We use Garsia–Procesi’s presentation of the cohomology of the Springer fiber, which describes the cohomology as a quotient of a polynomial ring analogous to the Borel construction of the cohomology of the flag variety [3]. Let \(R_n\) be the polynomial ring \(\mathbb {C}[x_1, \ldots , x_n]\). For each partition \(\lambda \) of n Tanisaki defined an ideal \(I_{\lambda } \subseteq R_n\) that is now called the Tanisaki ideal. To describe the Tanisaki ideal, we define certain sets of elementary symmetric functions.

Definition 5

Given a subset \(S \subseteq \{x_1, \ldots , x_n\}\) the elementary symmetric function \(e_i(S)\) is the polynomial

$$\begin{aligned} e_i(S) = \sum _{\begin{array}{c} T \text { such that }\\ T \subseteq S \text { and } |T|=i \end{array}} \prod _{x_j \in T} x_j. \end{aligned}$$

The set \(E_{i,j}^n\) is the set of elementary symmetric functions \(e_i(S)\) over all possible subsets \(S\subseteq \{x_1, \ldots , x_n\}\) of cardinality j.

Example 5

Let \(S = \{x_1, x_3, x_4\} \subseteq \{x_1, x_2, \ldots , x_5\}\). Then, \(e_2(S)\) is the polynomial \(x_1x_3 + x_1x_4 + x_3x_4\). Letting S run over all \(\left( {\begin{array}{c}5\\ 3\end{array}}\right) =10\) size three subsets of \(\{x_1, x_2, \ldots , x_5\}\) gives the 10 elementary symmetric polynomials \(e_2(S)\) comprising the set \(E_{2,3}^5\).

We follow Biagioli, Faridi and Rosas’s construction of the Tanisaki ideal [1, Definition 3.4].

Definition 6

Let \(\lambda = (\lambda _1, \ldots , \lambda _k)\) be a partition of n. The BFR-filling of \(\lambda \) is constructed as follows. From the leftmost column of \(\lambda \) to the rightmost column, place the numbers \(1, 2, \ldots , n-\lambda _1\) bottom to top skipping the top row. Finally, fill the top row from right to left with the remaining numbers \(n-\lambda _1+1, \ldots , n\). The BFR-generators are polynomials in the set \(\mathcal {G}(\lambda )\) defined as the following union: If the box filled with i has j in the top row of its column, then include the elements of \(E_{i,j}^n\) in the set \(\mathcal {G}(\lambda )\).

Example 6

Consider the partition \(\lambda = (4,2,2,1)\). Then, the BFR-filling of \(\lambda \) is

and hence, the set \(\mathcal {G}(\lambda )\) is

$$\begin{aligned} \mathcal {G}(\lambda ) = \bigcup _{i \in \{1,2,3,9\}} \!\!\! E^9_{i,9} \; \cup \; \bigcup _{i \in \{4,5,8\}} E^9_{i,8} \; \cup \; E^9_{7,7} \; \cup \; E^9_{6,6}. \end{aligned}$$

The first union of sets arises from the first column, the second from the second, and so on. Each set of the form \(E^9_{i,9}\) contains the single elementary symmetric function \(e_i(x_1, x_2, \ldots , x_9)\).

Theorem 1

(Biagioli–Faridi–Rosas  [1, Corollary 3.11]) Given a partition \(\lambda \) of n the Tanisaki ideal \(I_\lambda \) is generated by the set \(\mathcal {G}(\lambda )\).

3.2 Combinatorial presentation for the Springer representation and the Garsia–Procesi basis

It turns out that the natural \(S_n\)-action on \(R_n\) given by

$$\begin{aligned} w \cdot p(x_1, \ldots , x_n) = p(x_{w(1)}, \ldots , x_{w(n)}) \end{aligned}$$

restricts to the Tanisaki ideal \(I_{\lambda }\). Thus, the \(S_n\)-action on \(R_n\) induces an \(S_n\)-action on \(R_n/I_{\lambda }\). The key result of Garsia–Procesi’s work (with others [10, 21, 29]) is that this quotient is the cohomology of the Springer variety.

Proposition 2

(Garsia–Procesi [16]) Let \(R_n = \mathbb {C}[x_1, \ldots , x_n]\) and let \(I_\lambda \) be the Tanisaki ideal corresponding to the partition \(\lambda \). The quotient \(R_n/I_{\lambda }\) is isomorphic to the cohomology of the Springer variety \(H^*(Spr_{\lambda })\) as a graded \(S_n\)-representation.

Moreover, Garsia–Procesi describe an algorithm to compute a nice basis \(\mathcal {B}(\lambda )\) of monomials for \(R_n/I_{\lambda }\). Our exposition owes much to the presentation in the first author’s work [23, Definition 2.3.2].

Definition 7

(Garsia–Procesi [16, Sect. 1]) If \(\lambda = (1)\) is the unique partition of 1, then \(\mathcal {B}(\lambda ) = \{1\}\). If \(n>1\) and \(\lambda \) is a partition of n with k parts then:

  • Number the rightmost box in the ith row of \(\lambda \) with i for each \(i \in \{1,\ldots ,k\}\).

  • For each \(i \in \{1,\ldots ,k\}\) construct the partition \(\lambda _i\) of \(n-1\) by erasing the box labeled i and rearranging rows if needed to obtain a Young diagram once again.

  • Recursively define \(\mathcal {B}(\lambda )\) as

    $$\begin{aligned} \mathcal {B}(\lambda ) = \bigcup _{i=1}^k x_n^{i-1} \, \mathcal {B}(\lambda _i). \end{aligned}$$

From the GP-algorithm above, we construct the GP-tree as follows. Let \(\lambda \) sit alone at Level n in a rooted tree directed down. Since \(\lambda \) has k parts, create k edges labeled \(x_n^0, x_n^1, \ldots , x_n^{k-1}\) left-to-right to the k subdiagrams \(\lambda _i\) for each \(i \in \{1, \ldots , k\}\) in Level \(n-1\). For each of these subdiagrams and their descendants, recursively repeat this process. The process ends at Level 1, whose diagrams all contain one single box. Multiplying the edge labels on any downward path gives a unique GP-monomial in the GP-basis.

Example 7

We illustrate the first two steps of the GP-algorithm on the partition of 5 given by \(\lambda = (2,2,1)\). Number the far-right boxes and branch down from Level 5 to Level 4 of the recursion as follows:

After rearranging rows to obtain a Young diagram, we begin the recursion again on each of the three partitions to produce Level 3 of the tree.

By Level 1, there will be 30 diagrams, each equal to the partition (1). Recovering the Garsia–Procesi basis from this tree is equivalent to multiplying the edge labels of the 30 paths. The reader can verify that we obtain the following basis \(\mathcal {B}(\lambda )\):

Degree

#

Monomials in \(\mathcal {B}(\lambda )\)

0

1

1

1

4

\(x_i\) for \(2 \le i \le 5\)

2

9

\(x_i^2\) for \(3 \le i \le 5\) and \(x_ix_j\) for \(2 \le i<j \le 5\)

3

11

\(x_i x_j x_5\) for \(2 \le i<j \le 4\),

  

\(x_i x_j^2\) for \(2 \le i < j \le 5\), and

  

\(x_i^2 x_5\) for \(3 \le i \le 4\)

4

5

\(x_i x_j^2 x_5\) for \(2 \le i<j \le 4\) and \(x_i x_4 x_5^2\) for \(2 \le i \le 3\)

Garsia–Procesi bases have several nice containment properties that we use when analyzing the FI- and co-FI-structure of Springer representations. The first describes the relationship between dominance order and the Garsia–Procesi bases.

Proposition 3

(Garsia–Procesi [16, Proposition 4.1]) Let \(\lambda \) and \(\lambda '\) be partitions of n and suppose that \(\lambda \unlhd \lambda '\) in dominance order, namely that for all \(i \ge 1\) we have \(\lambda _1+\lambda _2+\cdots +\lambda _i \le \lambda '_1+\lambda '_2+\cdots +\lambda '_i\). Then \(\mathcal {B}(\lambda ') \subseteq \mathcal {B}(\lambda )\).

The second describes the relationship between containment of Young diagrams and the Garsia–Procesi bases.

Lemma 2

If \(\lambda \subseteq \lambda '\) then \(\mathcal {B}(\lambda ) \subseteq \mathcal {B}(\lambda ')\).

Proof

We prove the claim assuming that \(\lambda '\) has exactly one more box than \(\lambda \). Repeating the argument gives the desired result.

Consider the subdiagrams \(\lambda _1, \lambda _2, \ldots , \lambda _k\) obtained from \(\lambda '\) in the recursive definition of the Garsia–Procesi algorithm. The Garsia–Procesi algorithm says

$$\begin{aligned} \mathcal {B}(\lambda ') = \bigcup _{i=1}^k x_n^{i-1} \, \mathcal {B}(\lambda _i) = \mathcal {B}(\lambda _1) \cup \bigcup _{i=2}^k x_n^{i-1} \, \mathcal {B}(\lambda _i) \end{aligned}$$

so \(\mathcal {B}(\lambda ') \supseteq \mathcal {B}(\lambda _1)\). By construction \(\lambda _i\) is obtained from \(\lambda '\) by removing a box that is above and possibly to the right of the box removed for \(\lambda _{i+1}\) for each \(i \in \{1,\ldots ,k-1\}\). In particular \(\lambda _i \unlhd \lambda _{i+1}\) in dominance order. Proposition 3 implies that \(\mathcal {B}(\lambda _{i+1}) \subseteq \mathcal {B}(\lambda _i)\) for each i and so \(\mathcal {B}(\lambda ') \supseteq \mathcal {B}(\lambda _i)\) for each i. Since \(\lambda \) is obtained from \(\lambda '\) by removing a single box, we know \(\lambda = \lambda _i\) for some i. The claim follows. \(\square \)

4 The Springer representations with FI-module structure

Recall that the sequence of polynomial rings carries an FI-module structure and that both the Tanisaki ideal and the quotients \(R_n/I_{\lambda }\) carry an \(S_n\)-action. The question in this section is: do these fit together to give an FI-module structure on Springer representations? Lemma 2 suggests that the answer could be yes, since it proved that if \(\lambda \subseteq \lambda '\) then the Garsia–Procesi basis for \(R_n/I_{\lambda }\) is contained in the Garsia–Procesi basis for \(R_{n'}/I_{\lambda '}\).

We prove that this is misleading: The inclusion \(R_n/I_{\lambda } \hookrightarrow R_{n'}/I_{\lambda '}\) in no way preserves the \(S_n\) action (and is not what Church and Farb call a consistent sequence [9, pg.6]). In particular, we prove that there is no FI-module structure on sequences of Springer representations, except for the trivial representation. Church, Ellenberg, and Farb observed that the sequence of ideals of symmetric functions with no constant term is not an FI-ideal (see Example 3); in our language, they study the special case of the sequence of Tanisaki ideals \(\{I_{\lambda _1}, I_{\lambda _2}, \ldots \}\) where each \(\lambda _i\) is a column with i boxes.

Theorem 2

For each positive integer n, let \(\lambda _n\) denote a Young diagram with n boxes. The only sequence of Young diagrams \(\lambda _1 \subseteq \lambda _2 \subseteq \lambda _3 \subseteq \cdots \) for which the sequence of Tanisaki ideals \(\{I_{\lambda _1}, I_{\lambda _2}, I_{\lambda _3}, \ldots \}\) forms an FI-ideal is

figure a

.

Proof

Suppose that \(\lambda \) is a partition of \(n-1\) and \(\lambda '\) is a partition of n with \(\lambda \subseteq \lambda '\). We prove that if \(f: \{1,\ldots ,n-1\} \rightarrow \{1,\ldots ,n\}\) is an injection for which \(f_*(I_\lambda ) \subseteq I_{\lambda '}\) then \(I_{\lambda '} = \langle x_1, x_2, \ldots , x_n\rangle \). By definition of FI-modules this suffices to prove our claim.

We first show that \(I_{\lambda '}\) contains one of the variables \(x_1, \ldots , x_n\). Construct the BFR-fillings of both \(\lambda \) and \(\lambda '\) and compare the boxes labeled 1. A Young diagram with at least two rows has label 1 in the bottom box of the first column. A Young diagram with only one row has label 1 in the rightmost box of its row. If either \(\lambda \) or \(\lambda '\) has just one row, then the generating set for its Tanisaki ideal contains \(e_1(x_1)\). Hence, \(I_{\lambda '} \supseteq I_{\lambda }\) contains the variable \(x_1\). Otherwise the top-left boxes of the BFR-fillings for \(\lambda \) and \(\lambda '\) have the labels \(n-1\) and n, respectively, so \(\mathcal {G}(\lambda )\) contains \(e_1(x_1, \ldots , x_{n-1})\) and \(\mathcal {G}(\lambda ')\) contains \(e_1(x_1,\ldots ,x_n)\). In this case

$$\begin{aligned} e_1(x_1,\ldots ,x_n) - f_*(e_1(x_1,\ldots , x_{n-1})) = x_j \end{aligned}$$

lies in \(I_{\lambda '}\) for the unique \(j \in \{1,2,\ldots ,n\} - \text {Im}(f)\).

Thus, \(I_{\lambda '}\) contains at least one of the variables \(x_1, \ldots , x_n\). The Tanisaki ideal is invariant under the action of \(S_n\) that permutes the variables \(x_1, \ldots , x_n\) so in fact \(I_{\lambda '} = \left\langle x_1, \ldots , x_n \right\rangle \).

We conclude that

$$\begin{aligned} R_n/I_{\lambda '} = R_n/\left\langle x_1, \ldots , x_n \right\rangle \cong \mathbb {C}\end{aligned}$$

is trivial. Since \(\lambda '\) gives the trivial representation, it consists of a single row, as desired. \(\square \)

5 The Springer representation with the co-FI-module structure

In this section, we prove that Springer representations admit a co-FI-module structure. Recall from Sect. 2 that the sequence of polynomial rings forms a co-FI-module. This section shows that the natural restriction maps from \(k[x_1, \ldots , x_n]\) to \(k[x_1, \ldots , x_{n-1}]\) are defined on Tanisaki ideals, too. Our proof mimics a similar proof for the cohomology of flag varieties in the arXiv version of a paper by Church, Ellenberg, and Farb [7, Theorem 3.4]. A surprising feature of our result is that this co-FI-module structure exists for Springer representations corresponding to every possible sequence \(\lambda _1 \subseteq \lambda _2 \subseteq \lambda _3 \subseteq \cdots \) of Young diagrams.

The main features of the proof were outlined in previous sections, especially Sect. 2, which collected steps that reduce the proof that a sequence of Springer representations form a co-FI-module to proving that particular inclusions \(\iota _n^*: R_n \rightarrow R_{n-1}\) preserve Tanisaki ideals. We prove the main theorem first and then prove the key lemma.

Theorem 3

Suppose \(\lambda _1 \subseteq \lambda _2 \subseteq \cdots \) is a sequence of Young diagrams for which \(\lambda _n\) has n boxes for each n. Then, the sequence \(\{R_n/I_{\lambda _n}\}\) forms a graded co-FI-module of finite type.

Proof

Definition 3 states that to show \(\{R_n/I_{\lambda _n}\}\) is a graded co-FI-module of finite type, we must show 1) it is a graded co-FI-module and 2) its dual is a graded FI-module of finite type.

Assuming a field k of characteristic zero, Proposition 1 states that if R is a graded co-FI-module and I is a co-FI-ideal, then R/I is a graded co-FI-module. The sequence of polynomial rings is a graded co-FI-algebra under the co-FI-algebra structure that sends the map \(f: \{1,\ldots ,m\} \rightarrow \{1,\ldots ,n\}\) to the map with \(f^*(x_i)\) equal to \(x_{f^{-1}(i)}\) if \(i \le m\) and zero otherwise (see Example 4). Corollary 1 proves that if \(\{\lambda _1 \subseteq \lambda _2 \subseteq \cdots \}\) is a sequence of Young diagrams for which \(\iota _{n}^*(I_{\lambda _n}) \subseteq I_{\lambda _{n-1}}\) for every \(n \ge 2\), then the sequence of Tanisaki ideals \(\{I_{\lambda _1}, I_{\lambda _2}, \ldots \}\) forms a co-FI-ideal. Lemma 5 proves that \(\iota _{n}^*(I_{\lambda _n}) \subseteq I_{\lambda _{n-1}}\) for every \(n \ge 2\) and for every sequence of Young diagrams \(\{\lambda _1 \subseteq \lambda _2 \subseteq \cdots \}\). Thus, \(\{R_n/I_{\lambda _n}\}\) is a graded co-FI-module.

It is known that the dual \({ (R_n)^\vee }\) is a graded FI-module of finite type. The proof uses the facts that 0) the variables \(x_1, \ldots , x_n \in R_n\) form a basis for the dual to \(k^n\), 1) the jth graded part \((R_n)^\vee _j\) is isomorphic to the jth symmetric power of \(k^n\), and 2) the k-modules \(k, k^2, k^3, \ldots , k^n, \ldots \) are the parts of a natural FI-module M(1) whose algebraic properties are well understood; for details, see [8, proof of Theorem F].

Proposition 1 then implies that \(\{\left( R_n/I_{\lambda _n}\right) ^\vee \}\) is a graded FI-module of finite type, proving the claim. \(\square \)

5.1 Sequences of Tanisaki ideals form a co-FI-ideal

We now analyze the BFR-generators of the Tanisaki ideals to show that \(\iota _n^*(I_{\lambda '}) \subseteq I_{\lambda }\) when \(\lambda \subseteq \lambda '\). In an interesting twist, this co-FI-module structure exists for any path in the poset of Young diagrams. Our proof that these sequences of Tanisaki ideals \(\{I_{\lambda _i}\}\) form a co-FI-ideal occurs over the course of the following lemmas. Each result actually analyzes a subset of homogeneous polynomials of the same degree inside \(I_{\lambda }\) so our proofs preserve grading (though we do not use this fact).

We note the following well-known relation (e.g., [25, p. 21], [13, Equation (3.1)]).

Proposition 4

Let \(S \subseteq \{x_1,\ldots ,x_{n}\}\). If \(x_{n} \in S\) then \(e_i(S)\) decomposes as

$$\begin{aligned} e_i(S) = x_{n} \cdot e_{i-1}(S-\{x_{n}\}) + e_i(S-\{x_{n}\}). \end{aligned}$$

This immediately implies the following. (Recall that i is the degree of each function in the set \(E_{i,j}^n\) while j is the cardinality |S| of the variable subset \(S \subseteq \{x_1,x_2,\ldots ,x_n\}\).)

Lemma 3

$$\begin{aligned} \iota _n^*\left( E_{i,j+1}^{n}\right) = E_{i,j+1}^{n-1} \cup E_{i,j}^{n-1} \end{aligned}$$

where \(E_{i,k}^{n-1}\) is empty if \(k > n-1\) or if \(i > k\) or if \(n-1 < 1\).

The next lemma is our main tool: if an ideal contains \(E_{i,j}^n\) then it contains \(E_{i,j+1}^n\) as well.

Lemma 4

The \(\mathbb {Q}\)-linear span of \(E_{i,j}^n\) contains \(E_{i,j+1}^n\).

Proof

Let \(S' \subseteq \{x_1, \ldots , x_n\}\) be an arbitrary subset of cardinality \(j+1\). We construct \(e_i(S')\) explicitly in the \(\mathbb {Q}\)-linear span of \(E_{i,j}^n\). Consider the polynomial

$$\begin{aligned} p = \! \! \! \! \! \! \! \! \sum _{\begin{array}{c} S \text { such that } \\ S \subseteq S' \text { and } |S|=j \end{array}} \! \! \! \! \! \! \! \! e_i(S) \end{aligned}$$

which is in \(\langle E_{i,j}^n \rangle _{\mathbb {Q}}\) by definition. Let T be a subset of \(S'\) of cardinality i and consider the coefficient of the monomial \(\prod _{x_k \in T} x_k\) in p. Whenever \(S \subseteq S'\) contains T, the monomial \(\prod _{x_k \in T} x_k\) appears with coefficient 1 in \(e_i(S)\). There are \(j+1\) subsets of \(S'\) with cardinality j and all but i of them contain T. Hence, the coefficient of \(\prod _{x_k \in T} x_k\) in p is exactly \(j+1-i\). Thus, the polynomial \(\frac{1}{j+1-i}p = e_i(S')\) as desired. \(\square \)

Example 8

Consider the sets \(E_{2,3}^5\) and \(E_{2,4}^5\) and let \(S' = \{x_1,x_2,x_4,x_5\}\). The polynomial p from Lemma 4 is

$$\begin{aligned} p =&(x_1x_2+x_1x_4+x_2x_4) + (x_1x_2+x_1x_5+x_2x_5)\\&+(x_1x_4+x_1x_5+x_4x_5)+(x_2x_4+x_2x_5+x_4x_5) \end{aligned}$$

which simplifies as desired to

$$\begin{aligned} p = 2(x_1x_2+x_1x_4+x_1x_5+x_2x_4+x_2x_5+x_4x_5)=(4-2)e_2(S'). \end{aligned}$$

Applying Lemma 4 repeatedly gives the following.

Corollary 2

If an ideal \(I \subseteq \mathbb {Q}[x_1, \ldots , x_n]\) contains the subset \(E_{i,j}^n\) then it contains \(E_{i,j+k}^n\) for all \(k \in \{1, 2, \ldots , n-j\}\).

We now use these combinatorial properties of symmetric functions together with the BFR-generators of the Tanisaki ideal to prove the main lemma of this section

Lemma 5

Suppose that \(\lambda \) is a Young diagram with \(n-1\) boxes. If \(\lambda '\) is obtained from \(\lambda \) by adding one box then \(\iota _n^*(I_{\lambda '}) \subseteq I_{\lambda }\).

Proof

Below we give schematics for the relative configurations of \(\lambda \), \(\lambda '\), and the deleted box (shown in light gray).

We will compare the BFR-generators for \(I_{\lambda '}\) and \(I_{\lambda }\) for region A, region B, and the two different cases of gray boxes.

Each box in region A corresponds to BFR-generators \(E_{i,j}^{n-1}\) in \(I_{\lambda }\) and \(E_{i,j+1}^{n}\) in \(I_{\lambda '}\). In each of these cases, the image

$$\begin{aligned} \iota _n^*(E_{i,j+1}^{n}) = E_{i,j+1}^{n-1} \cup E_{i,j}^{n-1} \end{aligned}$$

by Lemma 3. By construction \(I_{\lambda }\) contains \(E_{i,j}^{n-1}\) so by Corollary 2 we know \(I_{\lambda }\) contains \(E_{i,j+1}^{n-1}\) as well. It follows that \(I_{\lambda }\) contains \(\iota _n^*(E_{i,j+1}^{n})\) for every box in region A.

If the box in \(\lambda ' - \lambda \) is not on the first row, then it is labeled \(E_{i,j+1}^{n}\) in \(I_{\lambda '}\) and the box above it is labeled \(E_{i,j}^{n-1}\) in \(I_{\lambda }\). This is the case of the previous paragraph.

If the box in \(\lambda ' - \lambda \) is on the first row, then \(I_{\lambda '}\) contains \(E_{i,i}^n\) while \(I_{\lambda }\) contains \(E_{i,i}^{n-1}\). Lemma 3 shows that \(\iota _n^*(E_{i,i}^{n}) = E_{i,i}^{n-1} \cup E_{i,i-1}^{n-1}\). However, \(E_{i,i-1}^{n-1}\) is empty by definition. So \(I_{\lambda }\) contains \(\iota ^*(E_{i,i}^{n})\) in this case too.

Finally, each box in region B corresponds to BFR-generators \(E_{i,j}^{n-1}\) in \(I_{\lambda }\) and \(E_{i+1,j+1}^{n}\) in \(I_{\lambda '}\). As above we know

$$\begin{aligned} \iota _n^*(E_{i+1,j+1}^{n}) = E_{i+1,j+1}^{n-1} \cup E_{i+1,j}^{n-1} \end{aligned}$$

by Lemma 3. The box labeled \(i+1\) in \(\lambda \) is either above but in the same column as in \(\lambda '\) or in a column to the right of \(i+1\) in \(\lambda '\). In either case \(E_{i+1,j-k}^{n-1}\) is a BFR-generator for \(\lambda \) for some nonnegative integer k. Corollary 2 implies that both \(E_{i+1,j+1}^{n-1}\) and \(E_{i+1,j}^{n-1}\) are in \(I_{\lambda }\). Hence, \(\iota _n^*(E_{i+1,j+1}^{n})\) is in \(I_{\lambda }\) for every box in region B, proving the claim. \(\square \)

6 Representation stability of Springer representations

We want to say that a sequence of representations \(\{V_n\}\) is representation stable if, when decomposed into irreducible representations, the sequence of multiplicities of each irreducible representation eventually becomes constant. This does not quite make sense because the irreducible representations of the symmetric group \(S_n\) depend on n, and in fact correspond to the partitions of n. The next definition describes a particular family of irreducible representations whose multiplicities we use to define representation stability.

Definition 8

(Irreducible \(S_n\)-representation \(V(\mu )_n\) [9, Sect. 2.1]) Let \(\mu {=} (\mu _1, \ldots , \mu _l)\) be a partition of k. For any \(n \ge k + \mu _1\) define the padded partition to be

$$\begin{aligned} \mu [n] := (n-k, \mu _1, \ldots , \mu _l). \end{aligned}$$

Define \(V(\mu )_n\) to be the irreducible \(S_n\)-representation

$$\begin{aligned} V(\mu )_n := V_{\mu [n]}. \end{aligned}$$

Note that every partition of n can be written as \(\mu [n]\) for a unique partition \(\mu \) and hence every irreducible \(S_n\)-representation is of the form \(V(\mu )_n\) for a unique partition \(\mu \).

We can now define representation stability precisely.

Definition 9

A sequence of \(S_n\)-representations \(V_1 {\mathop {\rightarrow }\limits ^{f_1}} V_2 {\mathop {\rightarrow }\limits ^{f_2}} V_3 {\mathop {\rightarrow }\limits ^{f_3}} \cdots \) is representation stable if

  1. 1.

    the linear maps \(f_n: V_n \rightarrow V_{n+1}\) are \(S_n\)-equivariant, in the sense that for each \(w \in S_n\) we have \(f_n \circ w = w(n+1) \circ f_n\) where \(w(n+1) \in S_{n+1}\) is the permutation that sends \(n+1 \mapsto n+1\) while otherwise acting as w;

  2. 2.

    the maps \(f_n: V_n \rightarrow V_{n+1}\) are injective;

  3. 3.

    the span of the \(S_{n+1}\)-orbit of the image \(f_n(V_n)\) is all of \(V_{n+1}\); and

  4. 4.

    if \(V_n\) decomposes into irreducible representations as

    $$\begin{aligned} V_n = \bigoplus _\mu c_{\mu ,n} V(\mu )_n \end{aligned}$$

    then there exists N so that the multiplicities \(c_{\mu ,n}\) are independent of n for all \(n \ge N\).

If N is independent of \(\mu \), then \(V_1 {\mathop {\rightarrow }\limits ^{f_1}} V_2 {\mathop {\rightarrow }\limits ^{f_2}} V_3 {\mathop {\rightarrow }\limits ^{f_3}} \cdots \) is uniformly representation stable.

FI-module structure corresponds to a sequence of representations being uniformly representation stable. Moreover, the stability of the multiplicities \(c_{\mu ,n}\) corresponds to a kind of stabilization of the characters of the representations and thus the dimensions of the representations.

Since \(\{R_n/I_{\lambda _n}\}\) forms a co-FI-module, we will actually show that the dual \(\{\left( R_n/I_{\lambda _n}\right) ^\vee \}\) is representation stable. The term Springer representation is applied interchangeably to these two dual representations (as described in the Introduction–see Remark 1). Moreover, the key implications of representation stability (including dimensions and characters) apply to \(\{R_n/I_{\lambda _n}\}\) by virtue of applying to its dual.

More precisely we have the following.

Corollary 3

Suppose \(\lambda _1 \subseteq \lambda _2 \subseteq \lambda _3 \subseteq \cdots \) is a sequence of Young diagrams for which \(\lambda _n\) has n boxes for each n. Then, the following are all true:

  1. 1.

    Each graded part of the sequence \(\{\left( R_n/I_{\lambda _n}\right) ^\vee \}\) is uniformly representation stable.

  2. 2.

    For each k and each n let \(\chi _{k,n}\) be the character in the kth graded part of \(R_n/I_{\lambda _n}\). The sequence \((\chi _{k,1},\chi _{k,2},\chi _{k,3},...)\) is eventually polynomial in the sense of [6].

  3. 3.

    For each k and n let \(d_{k,n}\) be the dimension of the kth graded piece of \(R_n/I_{\lambda _n}\) as a complex vector space. Then, the sequence \((d_{k,1},d_{k,2},d_{k,3},\ldots )\) is eventually polynomial, in the sense that there is an integer \(s_k\) and polynomial \(p_k(n)\) in n so that for all \(n \ge s_k\) the dimensions \(d_{k,n}=p_k(n)\).

Proof

This follows immediately from corresponding results of Church, Ellenberg, and Farb. Part (1) follows from Theorem 3 together with the definition of representation stability (or, e.g., [6, Theorem 1.13]). Part (2) follows for \(\left( R_n/I_{\lambda _n}\right) ^\vee \) from [6, Theorem 1.5] and for the dual representation \(R_n/I_{\lambda _n}\) because the characters of complex representations of \(S_n\) respect the operation of taking duals. Part (3) is an easy corollary of Part (2) for \(R_n/I_{\lambda _n}\) by [6, Theorem 1.5] and for its dual because they have the same dimension as complex representations. \(\square \)

7 Combinatorial consequences

The results of Corollary 3 open up new combinatorial questions about Springer fibers. Representation stability guarantees that for any sequence of Young diagrams \(\{\lambda _n\}\), the kth degree of the cohomology of the corresponding Springer fibers stabilizes as a polynomial \(p_k(n)\). But what is this polynomial? For instance, given the sequence \(\lambda _1 \subseteq \lambda _2 \subseteq \lambda _3 \subseteq \cdots \) of Young diagrams:

  • Is there an explicit formula for the dimensions \(p_k(n)\) of the kth graded part of \(R_n/I_{\lambda _n}\)? What if the sequence contains a particular family of Young diagrams (e.g., hooks, two-row, two-column, etc.)?

  • What is the minimal integer \(s_k\) at which the dimension of \(R_n/I_{\lambda _n}\) becomes polynomial? Given k and a sequence of Young diagrams, can we find some \(s_k\) (not necessarily minimal) after which the dimension of \(R_n/I_{\lambda _n}\) is polynomial?

  • Can we show the Springer representations have polynomial dimension via the monomial bases \(\mathcal {B}(\lambda _n)\) from Sect. 3?

To the best of our knowledge, these are entirely new questions about Springer representations; they arise only because of the consequences of representation stability. Moreover, our preliminary answers to these questions suggest additional combinatorial structures that may undergird Springer representations. We give details and some concrete results in this section, followed in the next by (more) open questions.

7.1 Minimal monomials

Our first results suggest that the exponents that appear in the monomials within each \(\mathcal {B}(\lambda )\) are more rigid than previously thought.

Definition 10

(Monomial type) Let \(x^{\alpha }\) be a monomial with exponent vector \((\alpha _1, {\ldots }, \alpha _n)\). In other words, the exponent of the variable \(x_i\) in \(x^{\alpha }\) is \(\alpha _i\). Let \((\alpha _{i_1}, \ldots , \alpha _{i_r})\) be the nonzero exponents in the order in which they appear in \(x^{\alpha }\). This r-tuple is called the monomial type of \(x^{\alpha }\).

For instance, both \(x_2x_3^2x_4\) and \(x_3x_9^2x_{11}\) have monomial type (1, 2, 1).

The following result says that if \(x_I^{\alpha }\) is a monomial of a fixed monomial type in the GP-basis \(\mathcal {B}(\lambda )\), then increasing the indices of the variables lexicographically while preserving the monomial type produces another monomial in the GP-basis \(\mathcal {B}(\lambda )\).

Theorem 4

Fix a composition \(\alpha = (\alpha _1, \ldots , \alpha _k)\). For each ordered sequence \(I=(i_1,\ldots ,i_k)\) let \(x_I^{\alpha }\) denote the monomial \(\prod _{j=1}^k x_{i_j}^{\alpha _j}\). If \(x_I^{\alpha } \in \mathcal {B}(\lambda )\) and \(I'\) satisfies

$$\begin{aligned} I \le I' \le (n-k+1,\ldots ,n-1,n) \end{aligned}$$

in lexicographic order then \(x_{I'}^{\alpha } \in \mathcal {B}(\lambda )\).

Proof

First note that we may simply consider the case when I and \(I'\) differ by exactly one in exactly one entry. Indeed, for each j with \(1 \le j \le k+1\), define the sequences \(I_j = (i_1, i_2, \ldots , i_{j-1}, i'_j, i'_{j+1}, \ldots , i'_k)\). Note that \(I=I_{k+1}\) and \(I' = I_1\). Thus, it suffices to prove that if \(x_{I_{j+1}}^{\alpha } \in \mathcal {B}(\lambda )\) then \(x_{I_{j}}^{\alpha } \in \mathcal {B}(\lambda )\). Furthermore, we may assume that \(i'_j = i_j+1\) since repeating the argument would successively increment the index and imply the result for more general \(i'_j\). Thus, we prove the claim for \(I_j\) and \(I_{j+1}\) assuming that \(i'_j = i_j+1\).

The GP-algorithm proceeds the same for both monomials \(x_{I_j}^{\alpha }\) and \(x_{I_{j+1}}^{\alpha }\) in the variables \(x_n, x_{n-1}, \ldots , x_{i_j+2}\). Let \(\lambda ^{(j+2)}\) be the Young diagram left after those steps. Implementing the GP-algorithm for \(x_{I_{j+1}}^{\alpha }\) removes a box from the first row and then from the \(\alpha _j+1^{\mathrm {th}}\) row of \(\lambda ^{(j+2)}\), whereas for \(x_{I_j}^{\alpha }\) it removes a box from the \(\alpha _j+1^{\mathrm {th}}\) row and then from the first row. Call the resulting Young diagrams \(\lambda ^{(j+1)}\) and \(\lambda ^{(j)}\), respectively.

We now show that \(\lambda ^{(j+1)} \unrhd \lambda ^{(j)}\) in dominance order. If the first row of \(\lambda ^{(j+2)}\) is strictly larger than the \(\alpha _j+1^{\mathrm {th}}\) row of \(\lambda ^{(j+2)}\), then the result is clear, since then \(\lambda ^{(j+1)} = \lambda ^{(j)}\). If not, then removing a box from the first row of \(\lambda ^{(j+2)}\) leaves a Young diagram whose shape is constrained: at least the first \(\alpha _j\) rows have the same length and there is at least one row immediately after of length one less. Thus, \(\lambda ^{(j+1)} \unrhd \lambda ^{(j)}\) with equality in the case that exactly the first \(\alpha _j+1\) rows of \(\lambda ^{(j+2)}\) are the same length, and any subsequent rows are smaller.

By Proposition 3, we have \(\mathcal {B}(\lambda ^{(j+1)}) \subseteq \mathcal {B}(\lambda ^{(j)})\). Since \(x_{i_1}^{\alpha _1}x_{i_2}^{\alpha _2} \cdots x_{i_{j-1}}^{\alpha _{j-1}}\) is a GP-monomial in \(\mathcal {B}(\lambda ^{(j+1)})\) it is also a GP-monomial in \(\mathcal {B}(\lambda ^{(j)})\) so the monomial \(x_{I_j}^{\alpha }\) can be obtained from the GP-algorithm starting with the shape \(\lambda ^{(j+2)}\). Thus, \(x_{I_j}^{\alpha }\) is in the GP-basis for \(\mathcal {B}(\lambda )\) as desired, and the claim follows. \(\square \)

In fact we conjecture that there is a unique minimal monomial of each monomial type, in the following lexicographic sense.

Definition 11

(Minimal monomials) Fix a monomial type \(\alpha \) and a partition \(\lambda \) of n. The monomial \(x_I^{\alpha } \in \mathcal {B}(\lambda )\) is a minimal monomial of type \(\alpha \) if for any other \(x_{I'}^{\alpha } \in \mathcal {B}(\lambda )\) of the same monomial type, the index sets satisfy \(I \le I'\) in lexicographic order.

The next result proves that when \(\lambda \) has exactly two rows then there is a unique minimal monomial of each monomial type in \(\mathcal {B}(\lambda )\). The previous result then says that this minimal monomial produces all other monomials of that fixed monomial type within \(\mathcal {B}(\lambda )\).

Theorem 5

Fix a two-row Young diagram \(\lambda = (n-k,k)\). Then, the basis \(\mathcal {B}(\lambda )\) of \(R_{n}/I_\lambda \) has the unique minimal monomial set \(\left\{ \prod \nolimits _{j=1}^i x_{2j} \right\} _{i=1}^k\).

Proof

Since \(\lambda \) has two rows, the maximum exponent that appears in a monomial for \(\lambda \) is 1. Since \(\lambda \) has k boxes in the second row, the maximum degree in a monomial for \(\lambda \) is k. So every monomial type that appears in \(\mathcal {B}(\lambda )\) is the partition given by j copies of 1, where j is any integer with \(0 \le j \le k\).

We claim that the minimal monomial of degree j is \(x_2 x_4 \cdots x_{2j}\). To see this, let \(x_I \in \mathcal {B}(\lambda )\) have degree j and index set \(I = (i_1, \ldots , i_j)\). It suffices to show that \((2, 4, \ldots , 2j) \le I\) in lexicographic order, namely that \(i_r \ge 2r\) for each \(r \in \{1,\ldots ,j\}\). Recall that in the GP-algorithm, exactly one rightmost box in a row is removed when going down each level of the GP-tree. Going from Level i to Level \(i-1\), the box removed is in the first row of the Level i diagram if \(i \notin \{i_1, \ldots , i_j\}\) and the second row otherwise. After a box is removed from the first row, the rows of the resulting Level \(i-1\) diagram are switched if necessary to maintain the shape of a Young diagram.

Now suppose \(r \in \{1,\ldots ,j\}\). The path in the GP-tree from Level \(i_r\) to Level \(i_r - 1\) removes a box from the second row of the Level \(i_r\) diagram \(\lambda ^{(i_r)}\). The diagram \(\lambda ^{(i_r)}\) must have at least r boxes in the second row since the \(r-1\) factors \(x_{i_{r-1}}, x_{i_{r-2}}, \ldots , x_{i_1}\) remaining in \(x_I\) correspond to \(r-1\) more second-row boxes removed in the remaining \(i_r - 2\) levels of the GP-tree below Level \(i_r - 1\). Since all Level \(i_r\) diagrams have \(i_r\) boxes and \(\lambda ^{(i_r)}\) has at least r boxes in its second row, we conclude \(i_r \ge 2r\) as desired.

\(\square \)

We conjecture that for each \(\mathcal {B}(\lambda )\) and each monomial type that appears in \(\mathcal {B}(\lambda )\), there is a unique minimal monomial of that monomial type (see Sect. 8.1).

7.2 Polynomial dimension when the number of rows exceeds a certain minimum

We can explicitly compute the polynomial dimensions described in Corollary 3 in some cases, including when the Young diagram has “enough” rows. In this case, the dimensions in low degree coincide with the corresponding dimensions for the diagonal coinvariant algebra, as described in the next theorem. Kim’s version of stability coincides with ours in this case [20, Remark in Sect. 4].

Theorem 6

Let \(\dim _i(R_n/I_{\lambda _n})\) denote the dimension of the ith-degree part of \(R_n/I_{\lambda _n}\). Let \(\{ \lambda _n \}\) be a nested sequence of Young diagrams such that \(|\lambda _n| = n\) for all n. If for some N the diagram \(\lambda _N\) contains at least \(k+1\) rows, then agrees with the dimension of the ith-degree part of the diagonal coinvariant algebra for all \(i \le k\) and \(n \ge N\).

In particular, for all \(i \le k\) and \(n \ge N\) we have where \(p_i(n)\) is the polynomial that gives the number of permutations of the set \(\{1,2,\ldots ,n\}\) with exactly i inversions.

Proof

Let \(\mathrm {col}_n = (1,\ldots ,1)\) denote the column partition with exactly n parts. The diagonal coinvariant algebra is \(R_n/I_{\mathrm {col}_n}\). There is a unique monomial \(x_2 x_3^2 x_4^3 \cdots x_n^{n-1} \in \mathcal {B}(\mathrm {col}_n)\) of maximal degree; it is obtained by choosing the far-right edge at each level in the GP-tree. Garsia–Procesi proved that their basis elements form a lower-order ideal with respect to division [16, Proposition 4.2]. It follows that the other elements of \(\mathcal {B}(\mathrm {col}_n)\) are precisely the divisors of this maximal monomial, so

$$\begin{aligned} \mathcal {B}(\mathrm {col}_n) = \{ x_2^{\alpha _2} x_3^{\alpha _3} x_4^{\alpha _4} \cdots x_n^{\alpha _n} \; | \; 0 \le \alpha _j \le j-1 \}. \end{aligned}$$
(1)

Now suppose that \(\lambda _n\) has at least \(k+1\) rows. Then \(\lambda _n \supseteq \mathrm {col}_{k+1}\) and so \(\mathcal {B}(\lambda _n) \supseteq \mathcal {B}(\mathrm {col}_{k+1})\) by Lemma 2. By Theorem 4 we can lexicographically increase the subscripts of each monomial in \(\mathcal {B}(\mathrm {col}_{k+1})\) to produce monomials \(x_I^{\alpha }\) for each k-element subset \(I \subseteq \{x_2, \ldots , x_n\}\) and exponents \(\alpha \) which satisfy \(0 \le \alpha _j \le j-1\) for each \(j=1,2,\ldots ,k+1\). This proves that \(\mathcal {B}(\lambda _n)\) also contains all of those possible elements of degrees \(0, 1, \ldots , k\) in the variable set \(\{x_2, \ldots , x_n\}\). Trivially, these are the same as the monomials of degrees \(0, 1, \ldots , k\) in the basis for \(\mathcal {B}(\mathrm {col}_n)\) above, so in these degrees the basis elements for \(R_n/I_{\lambda _n}\) and the diagonal coinvariant algebra \(R_n / I_{\mathrm {col}_n}\) coincide. The claim about \(p_i(n)\) follows from the similar statement for the ith graded part of \(R_n / I_{\mathrm {col}_n}\). \(\square \)

The polynomial function \(p_i(n)\) can be found explicitly as an alternating sum of certain combinations. Example 10 gives closed formulas for \(p_i(n)\) for the first few values of i.

Remark 3

Since the diagonal coinvariant algebra is the cohomology ring of the full flag variety, this result implies that the Garsia–Procesi bases for the cohomology of the Springer fibers coincide with the cohomology of the full flag variety in many degrees. More precisely, if \(\mathcal {S}_{\lambda _n}\) denotes the Springer fiber corresponding to the nilpotent of Jordan type \(\lambda _n\), then we have an isomorphism \(H^i(\mathcal {S}_{\lambda _n}) \cong H^*(GL_n(\mathbb {C})/B)\) for all \(i \le k\) and \(n \ge N\), where k and N are as given in Theorem 6.

Example 9

Below we give the basis elements for \(\mathcal {B}(\mathrm {col}_5)\) in degrees 0 through 3 and the cardinalities of the kth degree parts for \(4 \le k \le 10\), using the unimodality of the sequence of dimensions for \(k\ge 6\).

Degree

#

Monomials in \(\mathcal {B}(\mathrm {col}_5)\)

0

1

1

1

4

\(x_i\) for \(2 \le i \le 5\)

2

9

\(x_i^2\) for \(3 \le i \le 5\)

  

and \(x_ix_j\) for \(2 \le i<j \le 5\)

3

15

\(x_i x_j x_k\) for \(2 \le i<j<k \le 5\),

  

\(x_i x_j^2\) for \(2 \le i < j \le 5\),

  

\(x_i^2 x_j\) for \(3 \le i < j \le 5\), and

  

\(x_i^3\) for \(4 \le i \le 5\)

4

20

 

5

22

 

\(6 \le k \le 10\)

same cardinality as

 
 

\((10-k)^{\mathrm {th}} \text{ degree } \text{ part }\)

 

\(k > 10\)

0

 

Compare this to the basis \(\mathcal {B}(\lambda )\) for the partition \(\lambda = (2,2,1)\) that we computed in Example 7. The basis elements for \(\mathcal {B}(\lambda )\) and \(\mathcal {B}(\mathrm {col}_n)\) coincide in degrees 0, 1, and 2, as expected. However, in degree 3, the set \(\mathcal {B}(\mathrm {col}_5)\) contains four basis elements that are not in \(\mathcal {B}(\lambda )\), namely \(x_2 x_3 x_4\), \(x_3^2 x_4\), \(x_4^3\), and \(x_5^3\).

Example 10

We give the following closed formulas for the dimensions \(p_i(n)\) from Theorem 6 when \(0 \le i \le 6\):

  • When \(i=0\) we have \(p_0(n) = 1\).

  • When \(i = 1, 2, 3, 4\) we have

    $$\begin{aligned} p_i(n) = \left( {\begin{array}{c}n-2+i\\ i\end{array}}\right) - \left( {\begin{array}{c}n-3+i\\ i-2\end{array}}\right) . \end{aligned}$$
    (2)
  • When \(i=5\) we have

    $$\begin{aligned} p_5(n) = \left( {\begin{array}{c}n+3\\ 5\end{array}}\right) - \left( {\begin{array}{c}n+2\\ 3\end{array}}\right) + 1. \end{aligned}$$
  • When \(i=6\) we have

    $$\begin{aligned} p_6(n) = \left( {\begin{array}{c}n+4\\ 6\end{array}}\right) - \left( {\begin{array}{c}n+3\\ 4\end{array}}\right) + n. \end{aligned}$$

To prove these, let \(i \in \{1,2,3,4\}\). Our convention is that \(0 \in \mathbb {N}\). Define the set

$$\begin{aligned} \widetilde{E_i}&= \left\{ (d_2,\ldots ,d_n)\in \mathbb {N}^{n-1} \; \bigg \vert \; \sum _{k=2}^n d_k = i \text{ and } d_k \le i \right\} \end{aligned}$$

of nonnegative integer solutions to the equation \(d_2 + \cdots + d_n = i\). We know \(|\widetilde{E_i}| = \left( {\begin{array}{c}(n-2)+i\\ i\end{array}}\right) \). Define \(E_i \subseteq \widetilde{E_i}\) to be the subset satisfying also \(d_k \le k-1\) for each k so that \(p_i(n) = |E_i|\). We now compute \(|E_i| = |\widetilde{E_i}| - |E_i^c|\).

For each m, define the set \(F_{m,i} \subseteq \widetilde{E_i}\) by the condition that \(d_m > m-1\). Observe that \(E_i^c = \textstyle \bigcup \limits _{m=2}^{n} F_{m,i}\). Moreover, the sets \(F_{m,i}\) are pairwise disjoint. Indeed, suppose there were \(m_1<m_2\) with an element \((d_2,\ldots ,d_n) \in F_{m_1,i} \cap F_{m_2,i}\). Then \(d_{m_1} > m_1-1 \ge 1\) and \(d_{m_2} > m_2-1 \ge 2\) and so \(d_2 + \cdots + d_n \ge 5\). This contradicts \(i \in \{1,2,3,4\}\).

So \(|E_i^c| = \textstyle \sum \limits _{m=2}^n|F_{m,i}|\). We now find \(|F_{m,i}|\). Since \(d_m > m-1\) we write \(d_m = m + d_m'\) for some \(d_m' \ge 0\) and then count instead the solutions to \(d_2 + \cdots + d_m' + \cdots + d_n = i-m\). Thus

$$\begin{aligned} |E_i^c| = \sum _{m=2}^n |F_{m,i}| = \sum _{m=2}^n \left( {\begin{array}{c}(n-2) + (i-m)\\ i-m\end{array}}\right) . \end{aligned}$$

As long as \(i \le n\) this sum has the form \(|E_i^c| = \cdots + \left( {\begin{array}{c}r_m + 2\\ 2\end{array}}\right) + \left( {\begin{array}{c}r_m + 1\\ 1\end{array}}\right) + \left( {\begin{array}{c}r_m\\ 0\end{array}}\right) \) for some \(r_m\). Using a combinatorial identity we write \(|E_i^c| = \left( {\begin{array}{c}(n-2) + (i-2) + 1\\ i-2\end{array}}\right) \). This gives the formula for \(p_i(n)\) in Equation (2).

When \(i=5\) and \(i=6\) we can use essentially the previous argument, since only a few pairs of \(F_{m,i}\) share elements. When \(i=5\) the sets \(F_{2,5}\) and \(F_{3,5}\) both contain \((2,3,0,0,0,\ldots ,0)\). Inclusion–Exclusion gives the desired formula. Similarly if \(i=6\) then the intersection \(F_{2,6} \cap F_{3,6}\) contains the two elements \((3,3,0,0,\ldots ,0)\) and \((2,4,0,0,\ldots ,0)\) as well as the \(n-3\) elements \((2,3,d_4,\ldots ,d_n)\) where exactly one of the remaining entries \(d_k\) is nonzero (and thus is 1). Additionally, the intersection \(F_{2,6} \cap F_{4,6}\) contains the unique element \((2,0,4,0,0,\ldots ,0)\). Inclusion–Exclusion gives the desired formula.

This process becomes more complicated when i is larger.

Kim gives equivalent but different versions of the formulas in Example 10 [20, Example 3.3].

8 Open questions

This section describes a set of questions that arise from our analysis of representation stability, including how to characterize the polynomials into which the dimensions stabilize, and how to describe the monomials in \(\mathcal {B}(\lambda )\).

8.1 Minimal monomials and monomial types

In Sect. 7.1 we discussed monomial types and minimal monomials. We pose several questions here.

Question 1

For each Young diagram \(\lambda \) and monomial type \(\alpha \), is there a unique monomial \(x_I^{\alpha } \in \mathcal {B}(\lambda )\) that is minimal with respect to lexicographic ordering, in the sense that if \(x_{I'}^{\alpha } \in \mathcal {B}(\lambda )\) then \(I \le I'\)?

All examples that we have computed do in fact have minimal monomials.

There are several related questions about how the set of monomial types that appear in \(\mathcal {B}(\lambda )\) are related to \(\lambda \). The first essentially asks how to predict the monomial types that appear for a given \(\lambda \), while the second asks which \(\mathcal {B}(\lambda )\) contain a given monomial type \(\alpha \).

Question 2

For each Young diagram \(\alpha \), let \(\mathcal {A}(\lambda )\) be the set of monomial types in \(\mathcal {B}(\lambda )\). Is there a quick algorithm to construct \(\mathcal {A}(\lambda )\) for arbitrary \(\lambda \)?

We know there is an algorithm to construct \(\mathcal {A}(\lambda )\), namely use the GP-algorithm to find all of \(\mathcal {B}(\lambda )\) and then identify which monomial types appear. The previous question is asking for a direct algorithm, or for characterizations of the monomial types that do or do not appear.

Question 3

Fix a monomial type \(\alpha \). For which \(\lambda \) is there a monomial of type \(\alpha \) in \(\mathcal {B}(\lambda )\)?

For instance, \(\mathcal {B}(\lambda )\) contains a monomial of type (k) if and only if \(\lambda \) has at least \(k+1\) rows.

8.2 Polynomial dimension for particular families of Young diagrams

In this section, we return to questions from Sect. 7.2 to ask for concrete formulas for dimensions of the Springer representations, and especially for the polynomials \(p_k(n)\) to which they stabilize.

Representation stability relies on sequences of diagrams while the literature on Springer fibers instead uses families of Young diagrams like two-row, two-column, hook, and so on. To study families of Young diagrams, we choose a sequence \(\{\lambda _n\}\) that characterizes particular shapes, as Example 11 does for two-column Young diagrams.

Question 4

For what families of Young diagrams \(\{\lambda _n\}\) can we find an explicit closed formula for the polynomial dimension \(p_k(n)\)?

For instance, Kim gave a formula for two-row Young diagrams [19], following ideas due to Fresse [14, Example 4.5].

When we can identify minimal monomials (unique or not), strictly combinatorial techniques might then give information about the polynomial dimension formula. More precisely:

Question 5

Can we find minimal monomials for other shapes, and use them together with the index-incrementation tools in Theorem 4 to describe the dimension polynomials, either partially or completely?

For instance, Theorem 5 gave the minimal monomials for the case of the two-row diagrams \(\lambda = (n-k,k)\). In that case, the previous question sketches an alternate proof of Kim’s result for two-row Young diagrams.

8.3 Stable shapes

The central question of this section is to determine the integer \(s_k\) after which the polynomial \(p_k(n)\) counts the degree-k monomials in \(\mathcal {B}(\lambda _n)\). More colloquially, when does the dimension stabilize?

We conjecture that there is a core shape such that as soon as \(\lambda _n\) contains the core shape, the dimension stabilizes. The following describes one possible choice of a core shape, though we do not believe that it is in general minimal.

Definition 12

Let \(\lambda _1 \subseteq \lambda _2 \subseteq \cdots \) be a sequence of Young diagrams with \(|\lambda _n| = n\) for each n. Denote by \(\tau _{k+1}\) the staircase partition \((k+1, k, \ldots ,1)\). The k-stable shape of \(\{ \lambda _n \}\) is the partition \((\cup _n \, \lambda _n) \cap \tau _{k+1}\).

Example 11

Let \(\{ \lambda _n \}\) be the nested sequence of Young diagrams defined as follows: , and . Then, \(\{ \lambda _n \}\) is the following

The 1-, 2- and 3-stable shapes are and , respectively.

Conjecture 1

Let \(\{ \lambda _i \}\) be a sequence of Young diagrams such that \(\lambda _1 \subseteq \lambda _2 \subseteq \cdots \) with \(|\lambda _i| = i\). Let N be the smallest integer for which \(\lambda _N\) contains the k-stable shape of \(\{ \lambda _i \}\). Then, there exists a polynomial \(p_k(n)\) that gives the dimension of the kth graded part of the Springer representation \(R_n/I_{\lambda _n}\) for all \(n \ge N\).

If true, this conjecture would provide more tools to explicitly compute \(p_k(n)\).