Copyright © 2012 Darius Bayegan and Megumi Harada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace of a GKM space; Harada and Tymoczko also prove that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of . First we define the dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type regular nilpotent Hessenberg and any type nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. The algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Insko. Second, in a special case of regular nilpotent Hessenberg varieties, we prove that our pinball outcome is poset-upper-triangular, and hence the corresponding classes form a (pt)-module basis for the S¹-equivariant cohomology ring of the Hessenberg variety.

1. Introduction

The purpose of this paper is to further develop the theory of poset pinball, a combinatorial game introduced in [1] for the purpose of computing in equivariant cohomology rings (all cohomology rings in this note are with coefficients), in certain cases of type nilpotent Hessenberg varieties. One of the main uses of poset pinball in [1] is to construct module bases for the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces [1, Definition  4.5]. In the context of this paper, the ambient GKM space is the flag variety equipped with the action of the diagonal subgroup of , and the GKM-compatible subspaces are the nilpotent Hessenberg varieties. It is well recorded in the literature (e.g., [2] and references therein) that GKM spaces often have geometrically and/or combinatorially natural module bases for their equivariant cohomology rings; the basis of equivariant Schubert classes for is a famous example. The results of this paper represent first steps towards the larger goal of using poset pinball to construct a similarly computationally effective and convenient module bases for a GKM-compatible subspace by exploiting the structure of the ambient GKM space.

We briefly recall the setting of our results. Let be a nilpotent operator. Let be a function satisfying for all and for all . The associated Hessenberg variety is then defined as the following subvariety of : Since we deal exclusively with type in this paper, henceforth we omit this phrase from our terminology. Two special cases of Hessenberg varieties are of particular interest in this paper: when is the principal nilpotent operator (in this case is called a regular nilpotent Hessenberg variety) and when is the identity function for all (in this case is called a nilpotent Springer variety and is sometimes denoted ). Hessenberg varieties arise in many areas of mathematics, including geometric representation theory [3–5], numerical analysis [6], mathematical physics [7, 8], combinatorics [9], and algebraic geometry [10, 11], so it is of interest to explicitly analyze their topology, for example, the structure of their (equivariant) cohomology rings. We do so through poset pinball and Schubert calculus techniques, as initiated and developed in [1, 12, 13] and briefly recalled below.

The following relationship between two group actions on the nilpotent Hessenberg variety and the flag variety, respectively, allows us to use the theory of GKM-compatible subspaces and poset pinball. There is a natural subgroup of the unitary diagonal matrices which acts on (defined precisely in Section 2). The group , the maximal torus of , acts on in the standard fashion. It turns out that the -fixed points are a subset of the -fixed points . Moreover, the inclusion of into and the inclusion of groups into then induce a natural ring homomorphism As mentioned above, it is well known in Schubert calculus that the equivariant Schubert classes are a computationally convenient -module basis for . We refer to the images in of the equivariant Schubert classes via the projection (1.2) as Hessenberg Schubert classes. Given this setup and following [1], the game of poset pinball uses the data of the fixed points (considered as a partially ordered set with respect to Bruhat order) and the subset to determine a set of rolldowns in . It is shown in [1] that, under certain circumstances (one of which is discussed in more detail below), such a set of rolldowns in turn specifies a subset of the Hessenberg Schubert classes which form a -module basis of . Thus poset pinball is an important tool for building computationally effective module bases for the equivariant cohomology of Hessenberg varieties. Indeed, the results of [13] accomplish precisely this goal—that is, of constructing a module basis via poset pinball techniques—in the special case of the Peterson variety, which is the regular nilpotent Hessenberg variety with Hessenberg function defined by for and . Exploiting this explicit module basis, in [13, Theorem  6.12] the second author and Tymoczko give a manifestly positive Monk formula for the product of a degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class, expressed as a -linear combination of Peterson Schubert classes. This is an example of equivariant Schubert calculus in the realm of Hessenberg varieties, and it is an open problem to generalize the results of [13] to a wider class of Hessenberg varieties.

We now describe our main results. First, we explain in detail an algorithm which we dub the dimension pair algorithm and which associates to each -fixed point a permutation in , which we call the rolldown of following terminology in [1] and denoted . In the special cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties, we show that the set can be interpreted as the result of a successful game of Betti pinball (in the sense of [1]). The main motivation for our construction is that a successful outcome of Betti pinball can, under some circumstances, produce a module basis for the associated equivariant cohomology ring (cf. [1, Section  4.3]). In this sense, our algorithm represents a significant step towards the construction of module bases for the equivariant cohomology rings of general nilpotent Hessenberg varieties, thus extending the theory developed in [1, 13]. Although we formulate our algorithm in terms of dimension pairs and permissible fillings following terminology of Mbirika [14], the essential idea comes from a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Erik Insko.

Second, for a specific case of a regular nilpotent Hessenberg variety which we call a 334-type Hessenberg variety, we prove that the set of rolldowns obtained from the dimension pair algorithm is in fact poset-upper-triangular in the sense of [1]. As shown in [1], this is one of the possible circumstances under which we can conclude that the corresponding set of Hessenberg Schubert classes forms a module basis for the -equivariant cohomology ring of the variety. Thus our result gives rise to a new family of examples of Hessenberg varieties (and GKM-compatible subspaces) for which poset pinball successfully produces explicit module bases. We mention that the dimension pair algorithm also produces module bases in a special case of Springer varieties [15]. Although we do not know whether the dimension pair algorithm always succeeds in producing module bases for the -equivariant cohomology rings for a general nilpotent Hessenberg variety, the evidence thus far is suggestive. We leave further investigation to future work.

We give a brief summary of the contents of this manuscript. In Section 2 we recall some definitions and constructions necessary for later statements. In Section 3.1 we describe the dimension pair algorithm and prove that the result of the algorithm satisfies the conditions to be the outcome of a successful game of Betti poset pinball in the special cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties. We briefly review in Section 3.2 the theory developed in [1] which show that if the rolldown set obtained from a successful game of Betti poset pinball also satisfies poset-upper-triangularity conditions, then it yields a module basis in equivariant cohomology. In Sections 4 and 5 we prove that the dimension pair algorithm produces a poset-upper-triangular module basis in a special class of regular nilpotent Hessenberg varieties which we call 334-type Hessenberg varieties. We close with some open questions in Section 6.

2. Background

We begin with necessary definitions and terminology for what follows. In Section 2.1 we recall the geometric objects and the group actions under consideration. In Section 2.2 we recall some combinatorial definitions associated to Young diagrams. We recall a bijection between Hessenberg fixed points and certain fillings of Young diagrams in Section 2.3. The discussion closely follows previous work (e.g., [1, 13] and also [16]) so we keep exposition brief.

2.1. Hessenberg Varieties, Highest Forms, and Fixed Points

By the flag variety we mean the homogeneous space which is also identified with A Hessenberg function is a function satisfying for all and for all . We frequently denote a Hessenberg function by listing its values in sequence, . Let be a linear operator. The Hessenberg variety is defined as the following subvariety of : If is nilpotent, we say is a nilpotent Hessenberg variety, and if is the principal nilpotent operator (i.e., has one Jordan block with eigenvalue 0), then is called a regular nilpotent Hessenberg variety. If is nilpotent and is the identity function for all , then is called a nilpotent Springer variety and often denoted . In this manuscript we study in some detail the regular nilpotent case, and as such sometimes notate as when is understood to be the standard principal nilpotent operator.

Suppose is a nilpotent matrix in standard Jordan canonical form. It turns out that for many of our statements below we must use a choice of conjugate of which is in highest form [16, Definition  4.2]. We recall the following.

Definition 2.1 (see [16, Definitions 4.1 and 4.2]). (i) Let be any matrix. We call the entry a pivot of if is nonzero and if all entries below and to its left vanish, that is, if and if . Moreover, given , define to be the row of if the entry is a pivot, and 0 otherwise.
(ii) Let be an upper-triangular nilpotent matrix. Then we say is in highest form if its pivots form a nondecreasing sequence, namely, .

We do not require the details of the theory of highest forms of linear operators; for the purposes of the present manuscript it suffices to remark firstly that when is the principal nilpotent matrix, then is already in highest form, and secondly that any nilpotent matrix can be conjugated by an appropriate permutation matrix so that is in highest form. However, the following observation will be relevant in Section 2.3.

Remark 2.2. In this manuscript we always assume that our highest form has been chosen in accordance to the recipe described by Tymoczko in [16, Section 4]. Since the precise method of this construction is not relevant for the rest of the present manuscript, we omit further explanation here. In the case when is principal nilpotent, we take since is already in highest form and this is the form chosen by Tymoczko in [16]. A more detailed discussion of highest forms as it pertains to poset pinball theory is in [15].

For details on the following facts we refer the reader to, for example, [1, 13, 16] and references therein. Let be an nilpotent matrix in Jordan canonical form and let denote a permutation matrix such that is in highest form. It is known and straightforward to show that the following subgroup of preserves for as above and any Hessenberg function : Here is the standard maximal torus of consisting of diagonal unitary matrices.

This implies that the conjugate circle subgroup preserves . By abuse of notation we will denote both circle subgroups by , since it is clear by context which is meant. The -fixed points of and are isolated and are a subset of the -fixed points of . Since the set of -fixed points may be identified with the Weyl group , and since (resp., ) is a subset of , any Hessenberg fixed point may be thought of as a permutation .

2.2. Permissible Fillings, Dimension Pairs, Lists of Top Parts, and Associated Permutations

Recall that there is a bijective correspondence between the set of conjugacy classes of nilpotent complex matrices and Young diagrams (we use English notation for Young diagrams) with boxes, given by associating to the Young diagram with row lengths the sizes of the Jordan blocks of listed in weakly decreasing order. We will use this bijection to often treat such and as the same data; we sometimes denote by the Young diagram given as above corresponding to a nilpotent .

For more details on the following see [14].

Definition 2.3. Let be a Young diagram with boxes. Let be a Hessenberg function. A filling of by the alphabet is an injective placing of the integers into the boxes of . A filling of is called a -permissible filling if for every horizontal adjacency

in the filling we have .

Remark 2.4. In this manuscript the and will frequently be understood by context. When there is no danger of confusion we simply refer to permissible fillings.

Example 2.5. Let . Suppose and . Then

Definition 2.6. Let be a Hessenberg function and a Young diagram with boxes. A pair is a dimension pair of an -permissible filling of if the following conditions hold:(1), (2) is either(i)below in the same column of , or(ii)anywhere in a column strictly to the left of the column of , and(3)if there exists a box with filling directly adjacent to the right of , then .

For a dimension pair of , we will refer to as the top part of the dimension pair.

Example 2.7. Let , be as in Example 2.5. The dimension pairs in the permissible filling

Given a permissible filling of , we follow [14] and denote by the set of dimension pairs of . For each integer with , let so is the number of times occurs as a top part in the set of dimension pairs of . From the definitions it follows that . We call the integral vector the list of top parts of .

To each such we associate a permutation in as follows. As a preliminary step, for each with define where denotes the simple transposition in and 1 denotes the identity permutation. Now define the association It is not difficult to see that (2.6) is a bijection between the set of integral vectors satisfying for all and the group . In fact the word given by (2.6) is a reduced word decomposition of and the count the number of inversions in with as the higher integer. The following simple fact will be used later.

Fact 1. Suppose , are both lists of top parts. Suppose further that for all , we have . Then in Bruhat order. This follows immediately from the definition  (2.6).

Example 2.8. Continuing with Examples 2.5 and 2.7, for the permissible filling

Example 2.9. Let be as in Example 2.5. The filling

2.3. Bijection between Fixed Points and Permissible Fillings

For nilpotent Hessenberg varieties, the -fixed points are in bijective correspondence with the set of permissible fillings of the Young diagram , as we now describe. We will use this correspondence in the formulation of our dimension pair algorithm.

Suppose is a Young diagram with boxes. We begin by defining a bijective correspondence between the set of all fillings (not necessarily permissible) of with permutations in . Given a filling, read the entries of the filling by reading along each column from the bottom to the top, starting with the leftmost column and proceeding to the rightmost column. The association is then given by interpreting the resulting word as the one-line notation of a permutation. For example, the filling(2.7) has associated permutation 641523. It is easily seen that this is a bijective correspondence. Given a filling of , we denote its associated permutation by .

Remark 2.10. In the case when is the principal nilpotent matrix, the corresponding Young diagram has only one row, so the above correspondence simply reads off the (one row of the) filling from left to right. In this case we abuse notation and denote by just . For instance, the permissible filling of in Example 2.9 has associated permutation 43215.
Now let denote the set of -permissible fillings of . Recall that elements in are viewed as permutations in via the identification . The next proposition follows from the definitions and some linear algebra. It is proven and discussed in more detail in [15], where the notation used is slightly different.

Proposition 2.11. Fix a positive integer. Let be a Hessenberg function and a Young diagram with boxes. Suppose is a nilpotent operator in highest form as chosen in [16] (cf. Remark 2.2) with . Let denote the associated nilpotent Hessenberg variety. Then the map from the -fixed points to the set of permissible fillings is well defined and is a bijection.

Remark 2.12. In the case when is the principal nilpotent matrix, is the Young diagram with only one row. Thus the map (2.9) above simplifies to where we abuse notation (cf. Remark 2.10) and denote by .

3. The Dimension Pair Algorithm for Betti Poset Pinball for Nilpotent Hessenberg Varieties

In this section we first explain the dimension pair algorithm which associates to any Hessenberg fixed point a permutation in . The name is due to the fact that the construction proceeds by computing dimension pairs in appropriate permissible fillings. We then interpret this algorithm as a method for choosing rolldowns associated to the Hessenberg fixed points in a game of Betti poset pinball in the sense of [1]. The algorithm makes sense for any nilpotent Hessenberg variety, so it is defined in that generality in Section 3.1. However, our proof that the algorithm produces a successful outcome of Betti poset pinball in the sense of [1] is only for the special cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties. In Section 3.2 we briefly recall the setup and necessary results of poset pinball which allow us to conclude that our poset pinball result yields an explicit module basis for equivariant cohomology.

3.1. The Dimension Pair Algorithm for Nilpotent Hessenberg Varieties

Let be a nilpotent matrix in highest form chosen as in Remark 2.2 and let . Let be a Hessenberg function and the corresponding nilpotent Hessenberg variety.

The definition of the dimension pair algorithm is pure combinatorics. It produces for each Hessenberg fixed point an element in . Following terminology of poset pinball, we denote this function by

Definition 3.1 (the dimension pair algorithm). We define as follows.(1)Let and let be its corresponding permissible filling as defined in (2.9).(2)Let be the set of dimension pairs in the permissible filling .(3)For each with , set as in (2.4) and define .(4)Define where is the permutation associated to the integer vector defined in (2.6).

Example 3.2. Let , be as in Example 2.5. The permutation is in , as can be checked. The associated permissible filling is

We next show that the rolldown function defined by the dimension pair algorithm above satisfies the conditions to be a successful outcome of Betti poset pinball as in [1] in certain cases of nilpotent Hessenberg varieties. The statement of one of the conditions requires advance knowledge of the Betti numbers of nilpotent Hessenberg varieties, for which we recall the following result (reformulated in our language) from [16].

Theorem 3.3 (see [16, Theorem  1.1]). Let be a nilpotent matrix in highest form chosen as in Remark 2.2 and let . Let be a Hessenberg function and let denote the corresponding nilpotent Hessenberg variety. There is a paving by (complex) affine cells of such that(i)the affine cells are in one-to-one correspondence with , and(ii)the (complex) dimension of the affine cell corresponding to a fixed point is

In particular, Theorem 3.3 implies that the odd Betti numbers of are 0, and the th even Betti number is precisely the number of fixed points in such that . Given the regular nilpotent Hessenberg variety , denote by its th Betti number, that is, We may now formulate the conditions that guarantee that is a successful outcome of Betti pinball. For more details we refer the reader to [1, Section 3]. It suffices to check the following(1) is injective,(2)for every , we have in Bruhat order, and(3)for every , , we have where denotes the Bruhat length of .

We prove each claim in turn. For the first assertion we restrict to two special cases of Hessenberg varieties.

Lemma 3.4. Suppose that is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the function is injective.

Proof. Since the association given in (2.6) is a bijection, it suffices to show that the map which sends a Hessenberg fixed point to the list of top parts of its associated permissible filling is injective. Mbirika shows that, in the cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties, there exists an inverse to this map (Mbirika works with monomials in variables constructed from the list of top parts, but this is equivalent data) [14, Section 3.2]. The result follows.

Lemma 3.5. For every , one has in Bruhat order.

Proof. Since Bruhat order is preserved under taking inverses, it suffices to prove that is Bruhat-less than . For any permutation , set and let . Then the association (2.6) applied to the vector recovers the permutation . By definition of and the definition of dimension pairs, the set is always a subset of the set of inversions of the permutation . From Fact 1 it follows that the permutation is Bruhat-less than as desired.

Lemma 3.6. Let be a nilpotent matrix in highest form chosen as in Remark 2.2 and let . Let be a Hessenberg function and the associated nilpotent Hessenberg variety. For every , , one has where denotes the Bruhat length of .

Proof. By construction, has a reduced word decomposition consisting of precisely simple transpositions. Hence its Bruhat length is . By Theorem 3.3, is precisely the number of fixed points with so the result follows.

The following is immediate from the above lemmas and the definition of Betti pinball given in [1, Section 3].

Proposition 3.7. Suppose that is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the association given by the dimension pair algorithm is a possible outcome of a successful game of Betti poset pinball played with ambient partially ordered set equipped with Bruhat order, rank function given by Bruhat length, initial subset , and target Betti numbers .

Remark 3.8. Lemmas 3.5 and 3.6 hold for general nilpotent and Hessenberg functions . Hence to prove that Proposition 3.7 holds for more general cases of nilpotent Hessenberg varieties, it suffices to check that the injectivity assertion (1) above holds. We do not know counterexamples where the injectivity fails. It would be of interest to clarify the situation for more general and .

3.2. Betti Pinball, Poset-Upper-Triangularity, and Module Bases

In the context of a GKM-compatible subspace of a GKM space [1, Definition  4.5], it is explained in [1, Section 4] that the outcome of a game of poset pinball may be interpreted as specifying a set of equivariant cohomology classes which, under additional conditions, yields a module basis for the equivariant cohomology of the GKM-compatible subspace. In this paper, the GKM space is the flag variety with the standard -action and the GKM-compatible subspace is with the -action specified above. Consider the -module basis for given by the equivariant Schubert classes . The dimension pair algorithm then specifies the set where for any the class is defined to be the image of under the natural projection map induced by the inclusion of groups and the -equivariant inclusion of spaces . We refer to the images as Hessenberg Schubert classes.

Following the methods of [1] we view and as subrings of We denote by , the value of the th coordinate in the direct sums above, for , or , , respectively. If for all , , then the set in is called poset-upper-triangular (with respect to the partial order on induced from Bruhat order) [1, Definition 2.3]. Finally, recall that the cohomology degree of an equivariant Schubert class (and hence also the corresponding Hessenberg Schubert class ) is .

The following is immediate from [1, Proposition  4.14] and the above discussion.

Proposition 3.9. Let be either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Let be the dimension pair algorithm defined above. Suppose (3.11) holds for all . Then the Hessenberg Schubert classes form a -module basis for the -equivariant cohomology ring .

Therefore, in order to prove that the Hessenberg Schubert classes above form a module basis as desired, it suffices to show that they satisfy the upper-triangularity conditions (3.11) for all , . The proof of this assertion, for a special class of regular nilpotent Hessenberg varieties closely related to Peterson varieties, is the content of Sections 4 and 5.

We close the section with a brief discussion of matchings. Following [1, Section 4.3], define to be the (complex) dimension of the affine cell containing the fixed point in Tymoczko’s paving by affines of in Theorem 3.3. Then from the discussion above we know and since the cohomology degree of is , we see that the association from is also a matching in the sense of [1] with respect to and rank function on given by Bruhat length. Thus the fact that the form a module basis can also be deduced from [1, Theorem  4.18].

4. Poset-Upper-Triangularity of Rolldown Classes for 334-Type Hessenberg Varieties

In this section and in Section 5 we analyze in detail the dimension pair algorithm in the case of a Hessenberg variety which is closely related to the Peterson variety and in particular prove that the algorithm produces a poset-upper-triangular module basis for its -equivariant cohomology ring. Here and below the nilpotent operator under consideration is always the principal nilpotent, so we omit the from the notation and write . Similarly the corresponding Young diagram is always so we omit the from notation and write instead of .

We fix for this discussion the Hessenberg function given by The only difference between this function and the Hessenberg function for the Peterson variety studied in [13] is that the value of is 3 instead of 2. In this sense this is “close’’ to the Peterson case. Thus it is natural that much of our analysis follows that for Peterson varieties in [13], although it is still necessary to introduce new ideas and terminology to handle the Hessenberg fixed points in which do not arise in the Peterson case.

The Hessenberg function in (4.1) is trivial if since in that case which implies that the corresponding Hessenberv variety is equal to the full flag variety . Hence we assume throughout. Under this assumption and following the notation introduced in Section 2, the Hessenberg function is of the form . As such, for the purposes of this manuscript, we refer to this family of regular nilpotent Hessenberg varieties as 334-type Hessenberg varieties.

Our main result is the following theorem.

Theorem 4.1. Let and let be the 334-type Hessenberg variety in . Let be the dimension pair algorithm defined in Section 3. Then for all , . In particular the Hessenberg Schubert classes form a -module basis for the -equivariant cohomology ring .

For ease of exposition, and because the arguments required are of a somewhat different nature, we prove Theorem 4.1 by proving the two assertions in (4.2) separately, as follows.

Proposition 4.2. Let , , and be as above. Then, for all .

Proposition 4.3. Let , , and be as above. Then, for all , .

The proof of Proposition 4.2 is the content of Section 5. The main result of the present section is the upper-triangularity property asserted in Proposition 4.3. Its proof requires a number of preliminary results. We first begin by reformulating the problem in terms of Bruhat relations among the fixed points.

Lemma 4.4. Let , , and be as above. If for all , one has in Bruhat order, then the Hessenberg Schubert classes satisfy (4.4).

Proof. Recall that the equivariant Schubert classes are poset-upper-triangular with respect to Bruhat order on . In particular, for all , we have if . Since the Hessenberg Schubert classes are images of the Schubert classes and the diagram commutes, it follows that if for all , , we have in Bruhat order, then (4.4) follows.