Poset pinball, the dimension pair algorithm, and type A regular nilpotent Hessenberg varieties

In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. 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 the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for any type $A$ regular nilpotent Hessenberg and any type $A$ nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety $\Flags(\C^n)$. The definition of the algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Erik Insko. Second, in the special case of the type $A$ regular nilpotent Hessenberg varieties specified by the Hessenberg function $h(1)=h(2)=3$ and $h(i) = i+1$ for $3 \leq i \leq n-1$ and $h(n)=n$, we prove that the pinball result coming from the dimension pair algorithm is poset-upper-triangular; by results of Harada and Tymoczko this implies the corresponding equivariant cohomology classes form a $H^*_{S^1}(\pt)$-module basis for the $S^1$-equivariant cohomology ring of the Hessenberg variety.


INTRODUCTION
The purpose of this manuscript is to further develop the theory of poset pinball, a combinatorial game introduced in [10] for the purpose of computing in equivariant cohomology rings, 1 in certain cases of type A nilpotent Hessenberg varieties. One of the main uses of poset pinball in [10] is to construct module bases for the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces [10,Definition 4.5]. In the context of this manuscript, the ambient GKM space is the flag variety Fℓags(C n ) equipped with the action of the diagonal subgroup T of U (n, C), and the GKM-compatible subspaces are the nilpotent Hessenberg varieties. It is well-recorded in the literature (e.g. [18] 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 {σ w } w∈Sn for H * T (Fℓags(C n )) is a famous example. The results of this manuscript 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.
Date: December 30, 2010. 2000 Mathematics Subject Classification. Primary: 14M17; Secondary: 55N91. The second author is partially supported by an NSERC Discovery Grant, an NSERC University Faculty Award, and an Ontario Ministry of Research and Innovation Early Researcher Award. 1 All cohomology rings in this note are with C coefficients.
We briefly recall the setting of our results. Let N : C n → C n be a nilpotent operator. Let h : {1, 2, . . . , n} → {1, 2, . . . , n} be a function satisfying h(i) ≥ i for all 1 ≤ i ≤ n and h(i + 1) ≥ h(i) for all 1 ≤ i < n. The associated Hessenberg variety Hess(N, h) is then defined as the following subvariety of Fℓags(C n ): Since we deal exclusively with type A in this paper, henceforth we omit this phrase from our terminology. Two special cases of Hessenberg varieties are of particular interest in this manuscript: when N is the principal nilpotent operator (in this case Hess(N, h) is called a regular nilpotent Hessenberg variety) and when h is the identity function h(i) = i for all 1 ≤ i ≤ n (in this case Hess(N, h) is called a nilpotent Springer variety and is sometimes denoted S N ). Hessenberg varieties arise in many areas of mathematics, including geometric representation theory [9,15,16], numerical analysis [6], mathematical physics [12,14], combinatorics [8], and algebraic geometry [4,5], so it is of interest to explicitly analyze their topology, e.g. the structure of their (equivariant) cohomology rings. We do so through poset pinball and Schubert calculus techniques, as initiated and developed in [1,10,11] 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 S 1 subgroup of the unitary diagonal matrices T which acts on Hess(N, h) (defined precisely in Section 2). The group T , the maximal torus of U (n, C), acts on Fℓags(C n ) in the standard fashion. It turns out that the S 1 -fixed points Hess(N, h) S 1 are a subset of the T -fixed points Fℓags(C n ) T ∼ = S n . Moreover, the inclusion of Hess(N, h) into Fℓags(C n ) and the inclusion of groups S 1 into T then induces a natural ring homomorphism (1.1) H * T (Fℓags(C n )) → H * S 1 (Hess(N, h)). As mentioned above, it is well-known in Schubert calculus that the equivariant Schubert classes {σ w } w∈Sn are a computationally convenient H * T (pt)-module basis for H * T (Fℓags(C n )). We refer to the images in H * S 1 (Hess(N, h)) of the equivariant Schubert classes {σ w } w∈Sn via the projection (1.1) as Hessenberg Schubert classes. Given this setup and following [10], the game of poset pinball uses the data of the fixed points Fℓags(C n ) T ∼ = S n (considered as a partially ordered set with respect to Bruhat order) and the subset Hess(N, h) S 1 ⊆ Fℓags(C n ) T ∼ = S n to determine a set of rolldowns in S n . It is shown in [10] 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 H * S 1 (pt)-module basis of H * S 1 (Hess(N, h)). Thus poset pinball is an important tool for building computationally effective module bases for the equivariant cohomology of Hessenberg varieties. Indeed, the results of [11] accomplish precisely this goali.e. 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 h defined by h(i) = i + 1 for 1 ≤ i ≤ n − 1 and h(n) = n. Exploiting this explicit module basis, in [11,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 H * S 1 (pt)-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 [11] 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 S 1 -fixed point w ∈ Hess(N, h) S 1 a permutation in S n , which we call the rolldown of w following terminology in [10] and denoted roℓℓ(w) ∈ S n . In the special cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties, we show that the set {roℓℓ(w)} w∈Hess(N,h) S 1 can be interpreted as the result of a successful game of Betti pinball (in the sense of [10]). 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. [10,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 [10,11]. Although we formulate our algorithm in terms of dimension pairs and permissible fillings following terminology of Mbirika [13], 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 {roℓℓ(w)} w∈Hess(N,h) S 1 obtained from the dimension pair algorithm is in fact poset-upper-triangular in the sense of [10]. As shown in [10], 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 S 1 -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 [7]. Although we do not know whether the dimension pair algorithm always succeeds in producing module bases for the S 1 -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 [10] 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.
Acknowledgements. We thank Erik Insko for explaining to us the correspondence between the elements of an affine paving of regular nilpotent Hessenberg varieties and certain Schubert polynomials which motivates our dimension pair algorithm. We thank Barry Dewitt and Aba Mbirika for useful conversations and Rebecca Goldin for reviewing an initial draft of this manuscript and her many excellent suggestions for improving exposition. We are particularly indebted to Julianna Tymoczko for her ongoing support, for answering many questions, and for her suggestions on an earlier draft of this paper.

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. [10,11] and also [17]) so we keep exposition brief.
2.1. Hessenberg varieties, highest forms, and fixed points. By the flag variety we mean the homogeneous space GL(n, C)/B which is also identified with A Hessenberg function is a function h : {1, 2, . . . , n} → {1, 2, . . . , n} satisfying h(i) ≥ i for all 1 ≤ i ≤ n and h(i + 1) ≥ h(i) for all 1 ≤ i < n. We frequently denote a Hessenberg function by listing its values in sequence, h = (h(1), h(2), . . . , h(n) = n). Let N : C n → C n be a linear operator. The Hessenberg variety Hess(N, h) is defined as the following subvariety of Fℓags(C n ): If N is nilpotent, we say Hess(N, h) is a nilpotent Hessenberg variety, and if N is the principal nilpotent operator (i.e. has one Jordan block with eigenvalue 0), then Hess(N, h) is called a regular nilpotent Hessenberg variety. If N is nilpotent and h is the identity function h(i) = i for all 1 ≤ i ≤ n then Hess(N, h) is called a nilpotent Springer variety and often denoted S N . In this manuscript we study in some detail the regular nilpotent case, and as such sometimes notate Hess(N, h) as Hess(h) when N is understood to be the standard principal nilpotent operator. Suppose given N 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 N which is in highest form [17,Definition 4.2]. We recall the following. • Let X be any m × n matrix . We call the entry X ik a pivot of X if X ik is nonzero and if all entries below and to its left vanish, i.e., X ij = 0 if j < k and X jk = 0 if j > i. Moreover, given i, define r i to be the row of X ri,i if the entry is a pivot, and 0 otherwise. • Let N be an upper-triangular nilpotent n × n matrix. Then we say N is in highest form if its pivots form a nondecreasing sequence, namely r 1 ≤ r 2 ≤ · · · ≤ r n .
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 N is the principal nilpotent matrix then N is already in highest form, and secondly that any nilpotent matrix can be conjugated by an appropriate n × n permutation matrix σ so that N hf := σN σ −1 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 N hf = σN σ −1 has been chosen in accordance to the recipe described by Tymoczko in [17,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 N is principal nilpotent we take N hf = N since N is already in highest form and this is the form chosen by Tymoczko in [17]. A more detailed discussion of highest forms as it pertains to poset pinball theory is in [7].
For details on the following facts we refer the reader to e.g. [10,11,17] and references therein. Let N be an n × n nilpotent matrix in Jordan canonical form and let σ denote a permutation matrix such that N hf := σN σ −1 is in highest form. It is known and straightforward to show that the following S 1 subgroup of U (n, C) preserves Hess(N, h) for N as above and any Hessenberg function h: Here T n is the standard maximal torus of U (n, C) consisting of diagonal unitary matrices. This implies that the conjugate circle subgroup σS 1 σ −1 preserves Hess(N hf , h). By abuse of notation we will denote both circle subgroups by S 1 , since it is clear by context which is meant. The S 1 -fixed points of Hess(N, h) and Hess(N hf , h) are isolated, and are a subset of the T n -fixed points of Fℓags(C n ). Since the set of T n -fixed points Fℓags(C n ) T n may be identified with the Weyl group W = S n , and since Hess(N, h) S 1 (respectively Hess(N hf , h) S 1 ) is a subset of Fℓags(C n ) T n , any Hessenberg fixed point may be thought of as a permutation w ∈ S n .

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 n × n complex matrices N and Young diagrams 2 with n boxes, given by associating to N the Young diagram λ with row lengths the sizes of the Jordan blocks of N listed in weakly decreasing order. We will use this bijection to often treat such N and λ as the same data; we sometimes denote by λ N the Young diagram given as above corresponding to a nilpotent N .
For more details on the following see [13].  We denote a permissible filling of λ by T , in analogy with standard notation for Young tableaux. Next we focus attention on certain pairs of entries in a permissible filling T .   , 4). Note that (3,4) is not a dimension pair because 1 is directly to the right of the 3 and 4 ≤ h(1).
Given a permissible filling T of λ, we follow [13] and denote by DP T the set of dimension pairs of T . For each integer ℓ with 2 ≤ ℓ ≤ n, let (2.3) x ℓ := |{(a, ℓ) | (a, ℓ) ∈ DP T }| so x ℓ is the number of times ℓ occurs as a top part in the set of dimension pairs of T . From the definitions it follows that 0 ≤ x ℓ ≤ ℓ − 1 for all 2 ≤ ℓ ≤ n. We call the integral vector x = (x 2 , x 3 , . . . , x n ) the list of top parts of T .
To each such x we associate a permutation in S n as follows. As a preliminary step, for each ℓ with 2 ≤ ℓ ≤ n define where s i denotes the simple transposition (i, i + 1) in S n and 1 denotes the identity permutation. Now define the association It is not difficult to see that (2.4) is a bijection between the set of integral vectors x ∈ Z n−1 satisfying 0 ≤ x ℓ ≤ ℓ − 1 for all 2 ≤ ℓ ≤ n − 1 and the group S n . In fact the word given by (2.4) is a reduced word decomposition of ω(x) and the x ℓ count the number of inversions in ω(x) with ℓ as the higher integer. The following simple fact will be used later.   (2,3). Hence x = (1, 2, 1, 0) and the associated permutation ω(x) is s 1 (s 2 s 1 )s 3 .

Bijection between fixed points and permissible fillings.
For nilpotent Hessenberg varieties, the S 1fixed points Hess(N, h) S 1 are in bijective correspondence with the set of permissible fillings of the Young diagram λ = λ N , as we now describe. We will use this correspondence in the formulation of our dimension pair algorithm.
Suppose λ is a Young diagram with n boxes. We begin by defining a bijective correspondence between the set Fiℓℓ(λ) of all fillings (not necessarily permissible) of λ with permutations in S n . 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 Fiℓℓ(λ) ↔ S n is then given by interpreting the resulting word as the one-line notation of a permutation. For example the filling T = 1 2 3 4 5 6 has associated permutation 641523. It is easily seen that this is a bijective corresondence. Given a filling T of λ we denote its associated permutation by φ λ (T ). Remark 2.11. In the case when N is the principal nilpotent n × n matrix, the corresponding Young diagram λ = λ N = (n) 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 φ −1 λ (w) by just w. For instance, the permissible filling of λ = (5) in Example 2.10 has associated permutation 43215. Now let denote the set of (h, λ)-permissible fillings of λ. Recall that elements in Hess(N, h) S 1 are viewed as permutations in S n via the identification Fℓags(C n ) T n ∼ = S n . The next proposition follows from the definitions and some linear algebra. It is proven and discussed in more detail in [7], where the notation used is slightly different.

h) is well-defined and is a bijection.
Remark 2.13. In the case when N is the principal nilpotent n × n matrix, λ is the Young diagram with only one row. Thus the map (2.6) above simplifies to w → w −1 where we abuse notation (cf. Remark 2.11) and denote φ −1 λ (w −1 ) by w −1 .

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 S n . 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 [10]. 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 [10] 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. The definition of the dimension pair algorithm is pure combinatorics. It produces for each Hessenberg fixed point w ∈ Hess(N hf , h) S 1 an element in S n . Following terminology of poset pinball, we denote this function by roℓℓ : Hess(N hf , h) S 1 → S n .

Example 3.2.
Let λ, h be as in Example 2.5. The permutation w = 43215 ∈ S n is in Hess(N hf , h) S 1 , as can be checked. The associated permissible filling is 4 3 2 1 5 . In Example 2.10 we saw that the associated permutation is s 1 (s 2 s 1 )s 3 , so we conclude roℓℓ(w) = s 3 (s 1 s 2 )s 1 .
We next show that the rolldown function roℓℓ : Hess(h) S 1 → S n defined by the dimension pair algorithm above satisfies the conditions to be a successful outcome of Betti poset pinball as in [10] 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 [17]. • the affine cells are in one-to-one correspondence with Hess(N hf , h) S 1 , and • the (complex) dimension of the affine cell C w corresponding to a fixed point w ∈ Hess(N, h) S 1 is In particular, Theorem 3.3 implies that the odd Betti numbers of Hess(N hf , h) are 0, and the 2k-th even Betti number is precisely the number of fixed points w in We may now formulate the conditions that guarantee that roℓℓ : Hess(N hf , h) S 1 → S n is a successful outcome of Betti pinball. For more details we refer the reader to [10,Section 3]. It suffices to check the following: (1) roℓℓ : Hess(N hf , h) S 1 → S n is injective, (2) for every w ∈ Hess(N hf , h) S 1 , we have roℓℓ(w) ≤ w in Bruhat order, and (3) for every k ≥ 0, k ∈ Z, we have where ℓ(roℓℓ(w)) denotes the Bruhat length of roℓℓ(w) ∈ S n . We prove each claim in turn. For the first assertion we restrict to two special cases of Hessenberg varieties.

Lemma 3.4. Suppose that Hess(N hf , h) is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the function roℓℓ : Hess
is a bijection it suffices to show that the map which sends a Hessenberg fixed point w ∈ Hess(h) S 1 to the list of top parts x 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 n − 1 variables constructed from the list of top parts, but this is equivalent data) [13, Section 3.2]. The result follows.
Proof. Since Bruhat order is preserved under taking inverses, it suffices to prove that ω(x) is Bruhat-less than w −1 . For any permutation u ∈ S n , set y ℓ := {(a, ℓ) | (a, ℓ) is an inversion in u} and let y := (y 2 , y 3 , . . . , y n ). Then the association (2.4) applied to the vector y recovers the permutation u. By definition of φ λ and the definition of dimension pairs, the set DP φ −1 λ (w −1 ) is always a subset of the set of inversions of the permutation w −1 . From Fact 2.8 it follows that the permutation ω(x) is Bruhat-less than w −1 as desired.
Proof. By construction, roℓℓ(w) has a reduced word decomposition consisting of precisely The following is immediate from the above lemmas and the definition of Betti pinball given in [10, Section 3]. Proposition 3.7. Suppose that Hess(N hf , h) is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the association w → roℓℓ(w) given by the dimension pair algorithm is a possible outcome of a successful game of Betti poset pinball played with ambient partially ordered set S n equipped with Bruhat order, rank function ρ = ℓ : S n → Z given by Bruhat length, initial subset Hess(h) S 1 ⊆ S n , and target Betti numbers b k := dim C H 2k (Hess(h); C). Remark 3.8. Lemmas 3.5 and 3.6 hold for general nilpotent N hf and Hessenberg functions h. 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 N hf and h.

Betti pinball, poset-upper-triangularity, and module bases.
In the context of a GKM-compatible subspace of a GKM space [10,Definition 4.5], it is explained in [10, 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 Fℓags(C n ) with the standard T n -action and the GKM-compatible subspace is Hess(N hf , h) with the S 1 -action specified above. Consider the H * T n (pt)module basis for H * T n (Fℓags(C n )) given by the equivariant Schubert classes {σ w } w∈Sn . The dimension pair algorithm then specifies the set where for any u ∈ S n the class p u := π(σ u ) is defined to be the image of σ u under the natural projection map π : H * T n (Fℓags(C n )) → H * S 1 (Hess(N hf , h)) induced by the inclusion of groups S 1 ֒→ T n and the S 1 -equivariant inclusion of spaces Hess(N hf , h) ֒→ Fℓags(C n ). We refer to the images p u as Hessenberg Schubert classes.
Following the methods of [10] we view H * T n (Fℓags(C n )) and H * S 1 (Hess(N hf , h)) as subrings of We denote by σ w (w ′ ), p roℓℓ(w) (w ′ ) the value of the w ′ -th coordinate in the direct sums above, for w, w ′ ∈ S n or w, w ′ ∈ Hess(N hf , h) S 1 respectively. If (Hess(N hf , h)) is called poset-upper-triangular (with respect to the partial order on Hess(N hf , h) S 1 ⊆ S n induced from Bruhat order) [10, Definition 2.3]. Finally, recall that the cohomology degree of an equivariant Schubert class σ w (and hence also the corresponding Hessenberg Schubert class p w ) is 2 · ℓ(w).
The following is immediate from [10, Proposition 4.14] and the above discussion.  (Hess(N hf , h)). 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 (3.2) for all w, w ′ ∈ Hess(N hf , h) S 1 . 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 [10, Section 4.3], define to be the (complex) dimension of the affine cell C w containing the fixed point w in Tymoczko's paving by affines of Hess(N hf , h) in Theorem 3.3. Then from the discussion above we know and since the cohomology degree of p roℓℓ(w) is 2 · ℓ(roℓℓ(w)), we see that the association w → roℓℓ(w) from Hess(N hf , h) S 1 → S n is also a matching in the sense of [10] with respect to deg Hess(N hf ,h) and rank function ρ on S n given by Bruhat length. Thus the fact that the {p roℓℓ(w) | w ∈ Hess(N hf , h) S 1 } form a module basis can also be deduced from [10, Theorem 4.18].

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 S 1 -equivariant cohomology ring. Here and below the nilpotent operator N under consideration is always the principal nilpotent, so we omit the N from the notation and write Hess(h). Similarly the corresponding Young diagram is always λ = (n) so we omit the λ from notation and write PF iℓℓ(h) instead of PF iℓℓ(λ, h).
We fix for this discussion the Hessenberg function given by The only difference between this function h and the Hessenberg function for the Peterson variety studied in [11] is that the value of h(1) is 3 instead of 2. In this sense this h is "close" to the Peterson case. Thus it is natural that much of our analysis follows that for Peterson varieties in [11], although it is still necessary to introduce new ideas and terminology to handle the Hessenberg fixed points in Hess(h) S 1 which do not arise in the Peterson case.
which implies that the corresponding Hessenberv variety Hess(h) is equal to the full flag variety Fℓags(C 3 ). Hence we assume n ≥ 4 throughout. Under this assumption and following the notation introduced in Section 2, the Hessenberg function is of the form h = (3, 3, 4, · · · ). 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.
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.
Proof. Recall that the equivariant Schubert classes are poset-upper-triangular with respect to Bruhat order on S n . In particular, for all w, w ′ ∈ S n we have σ w (w ′ ) = 0 if w ′ ≥ w. Since the Hessenberg Schubert classes are images of the Schubert classes and the diagram commutes, it follows that if for all w, w ′ ∈ Hess(h) S 1 , we have in Bruhat order then (4.4) follows.
The rest of this section is devoted to the proof of (4.5), which by Lemma 4.4 then proves Proposition 4.3.
4.1. Fixed points and associated subsets for the 334-type Hessenberg variety. In this section we enumerate the fixed points in the 334-type Hessenberg variety and also associate to each fixed point in Hess(h) S 1 a subset of {1, 2, . . . , n − 1}. As we show below, the set of fixed points in the Peterson variety is a subset of the fixed points of the 334-type Hessenberg variety, so the main task is to describe the new fixed points which arise in the 334-type case. We begin with a general observation.

The inclusion Hess(h ′ ) ֒→ Hess(h) is S 1 -equivariant and in particular Hess
Proof. Let V • = (V i ) denote an element in Fℓags(C n ). By definition the regular nilpotent Hessenberg variety Hess(h ′ ) associated to h ′ is where N is the principal nilpotent operator. Since Applying Lemma 4.5 to the Hessenberg function corresponding to the Peterson variety Hess(h ′ ) and h the 334-type Hessenberg function (4.1), we conclude that all fixed points in Hess(h ′ ) S 1 also arise as fixed points in Hess(h) S 1 . We refer to the elements of Hess(h ′ ) S 1 (viewed as elements of Hess(h) S 1 ) as Peterson-type fixed points. It therefore remains to describe Hess(h) S 1 \ Hess(h ′ ) S 1 . It turns out to be convenient to do this by first describing PF iℓℓ(h) \ PF iℓℓ(h ′ ).
We first introduce some terminology. Given a permutation w = (w(1) w(2) · · · w(n)) in one-line notation and some i, ℓ, we say that the entries {w(i), w(i + 1), . . . , w(i + ℓ)} form a decreasing staircase, or simply a staircase, if w(j + 1) = w(j) − 1 for all i ≤ j < i + ℓ. For example for w = 4327516, the segment 432 is a staircase, but 751, though the entries decrease, is not. We will say that a consecutive series of staircases is an increasing sequence of staircases (or simply increasing staircases) if each entry in a given staircase is smaller than any entry in any following staircase (reading from left to right). For instance, w = 654987321 is a sequence of staircases 654, 987, and 321, but is not an increasing sequence of staircases since the entries 4, 5, 6 are not smaller than the entries in the later staircase 321. However, w = 321654987 is an increasing sequence of (three) staircases 321, 654, and 987.
It is shown in [11] that the S 1 -fixed points of the Peterson variety Hess(h ′ ) consist precisely of those permutations w ∈ S n such that the one-line notation of w is an increasing sequence of staircases. Since such w are equal to their own inverses, the permissible fillings PF iℓℓ(h ′ ) corresponding to Hess(h ′ ) are precisely those which are increasing sequences of staircases (cf. Remark 2.13). We now describe the permissible fillings PF iℓℓ(h) which are not Peterson-type fillings. We use the language of h-tableau trees introduced by Mbirika; see [13, Section 3.1] for definitions. Recall from Remark 2.11 that we identify permissible fillings with permutations in S n via one-line notation. Lemma 4.6. Let n ≥ 4 and let Hess(h) be the 334-type Hessenberg variety in Fℓags(C n ). Let w ∈ PF iℓℓ(h) be a permissible filling for Hess(h) which is not of Peterson type, i.e., w ∈ PF iℓℓ(h) \ PF iℓℓ(h ′ ). Then precisely one of the following hold: • The one-line notation of w is of the form where w ′ is a (possibly empty) staircase such that w ′ 3 is also a staircase, and w ′′ is an increasing sequence of staircases. We refer to these as 312-type permissible fillings. • The one-line notation of w is of the form where w ′ is a (possibly empty) staircase such that w ′ 3 is also a staircase, and w ′′ is an increasing sequence of staircases. We refer to these as 231-type permissible fillings. Moreover, any filling satisfying either of the above conditions appears in PF iℓℓ(h) \ PF iℓℓ(h ′ ). . In particular, since we saw above that the edges going down from Level 3 onwards are identical in both the Peterson and 334-type Hessenberg case, it follows that the branches of the tree emanating downwards from the two Level 3 vertices 3 2 1 •, 2 1 3 • (coming from • 2 • 1 •) and the two vertices 1 • 3 2 •, 1 2 • 3 • (coming from • 1 • 2 •) are identical to the corresponding branches in the h-tableau tree for the Peterson Hessenberg function. Hence all permissible fillings at the final Level n of these branches are of Peterson type. In contrast, the branches emanating from 2 • 3 1 • and • 3 1 2 • do not appear in the Peterson h-tableau tree, and none of the fillings appearing at Level n in these branches can be Peterson permissible fillings since a 3 appears directly before a 1. Hence it is precisely these branches which account for the permissible fillings which are not of Peterson type. As noted above, the rest of the branch only has 2 edges going down from each vertex with h-permissible positions determined exactly as in the Peterson case. In particular, except for the exceptional 3 appearing directly to the left of a 1, the fillings must consist of decreasing staircases and all possible arrangements of decreasing staircases do appear. The result follows. We now give explicit descriptions of the corresponding non-Peterson-type elements in Hess(h) S 1 , obtained by taking inverses of the permissible fillings described in Lemma 4.6.
Definition 4.8. Let w ∈ Hess(h) S 1 . We say w is a 312-type (respectively 231-type) fixed point if its inverse w −1 is a permissible filling of 312-type (respectively 231-type).
As observed above, since Peterson-type permissible fillings are equal to their own inverses, in that case there is no distinction between the fillings and their associated fixed points. For the 312 and 231-types, however, this is not the case. We record the following. The proof is a straightforward computation and is left to the reader. Lemma 4.9. Let w be a 312-type (respectively 231-type) permissible filling. Let a 2 be the integer such that a 2 + 1 is the first entry (respectively second entry) in the one-line notation of w. Let w −1 be the corresponding 312-type (respectively 231 type) fixed point. Then: • the one-line notation of w −1 is the same as that of w for all ℓ-th entries with ℓ > a 2 + 1, • if w is 312-type, then the first a 2 + 1 entries of the one-line notation of w −1 are (4.10) a 2 a 2 + 1 a 2 − 1 a 2 − 2 · · · 2 1 • if w is 231-type, then the first a 2 + 1 entries of the one-line notation of w −1 are (4.11) a 2 + 1 1 a 2 a 2 − 1 · · · 3 2 In the case of the Peterson variety, there is a convenient bijective correspondence between the set of S 1fixed points of the Peterson variety and subsets A of {1, 2, . . . , n − 1} given as follows [11,Section 2.3]. Let w be a Peterson-type fixed point. Then the corresponding subset is In the case of the 334-type Hessenber variety, it is also useful to assign a subset of {1, 2, . . . , n − 1} to each fixed point as follows.
Definition 4.10. Let w ∈ Hess(h) S 1 . The associated subset of {1, 2, . . . , n} corresponding to w, notated A(w), is defined as follows: • Suppose w is of Peterson type. Then A(w) is defined to be the set A in (4.12).
• Suppose w is 312-type. Consider the permutation w ′ := ws 1 (i.e. swap the a 2 and the a 2 + 1 in the one-line notation (4.10)). This is a fixed point of Peterson type. Define A(w) := A(w ′ ). • Suppose w is 231-type. Consider the permutation w ′ = ws 2 s 3 · · · s a2 (i.e. move the 1 to the right of the 2 in the one-line notation (4.11)). This is a fixed point of Peterson type. Define A(w) := A(w ′ ).  It is useful to observe that the 312-type and 231-type fixed points have associated subsets that always contain 1 and 2. Proof. From the explicit descriptions of the one-line notation of the 312 type (respectively 231-type) fixed points given above, we know that the initial segment a 2 a 2 + 1 · · · 2 1 (respectively a 2 + 1 1 a 2 · · · 3 2) in the one-line notation is such that a 2 ≥ 2. From Definition 4.10 it follows that the first decreasing staircase of the associated Peterson-type fixed point ws 1 (respectively ws 2 s 3 · · · s a2 ) is of length at least 3. In particular, the first staircase starts with an integer k which is ≥ 3. The result follows.
As noted in Remark 4.12, the association w → A(w) given in Definition 4.10 is not one-to-one and hence in particular not a bijective correspondence. This makes our analysis more complicated than in [11], but the notion is still useful for our arguments below.

Reduced word decompositions for 334-type fixed points and rolldowns.
In this section we fix particular choices of reduced word decompositions for the fixed points in Hess(h) S 1 which we use in our arguments below. We also compute, and fix choices of reduced words for, the rolldowns roℓℓ(w) of the fixed points.
The association w → A(w) of the previous section allows us to describe these reduced word decompositions in relation to that of the Peterson-type fixed points. Let a be a positive integer and k a non-negative integer. Recall that a reduced word decomposition of the maximal element (the full inversion) in the subgroup S {a,a+1,...,a+k+1} ⊆ S n is given by s a (s a+1 s a )(s a+2 s a+1 s a ) · · · (s a+k s a+k−1 · · · s a+1 s a ).
For the purposes of this manuscript, we call this the standard reduced word (decomposition) for the maximal element. (This is different from the choice of reduced word decomposition used in [ [1,3] w [5,6] = s 1 (s 2 s 1 )(s 3 s 2 s 1 )s 5 (s 6 s 5 ).
We now fix a reduced word decomposition of the non-Peterson-type fixed points.

Lemma 4.15. Let w ∈ Hess(h) S 1 be a fixed point which is not of Peterson type and let
be the associated subset with its decomposition into maximal consecutive substrings.
Proof. For the first assertion, observe that the explicit description of the one-line notation 312-type fixed points in (4.10) implies that w has precisely 1 fewer inversion than w A(w) . An explicit computation shows that the given word (4.17) is equal to w, so it is a word decomposition of w with exactly as many simple transpositions as the Bruhat length of w. In particular it must be reduced. A similar argument proves the second assertion. Henceforth we always use the reduced words given above. Next we explicitly describe the rolldowns roℓℓ(w) associated to each w in Hess(h) S 1 by the dimension pair algorithm. We begin with the Peterson-type fixed points. It turns out there are two important subcases of Peterson-type fixed points.

Definition 4.17.
We say that a Peterson-type fixed point w contains the string 321 (or simply contains 321) if, in the one-line notation of w, the string 321 appears (equivalently, if {1, 2} ⊆ A(w)). We say w does not contain the string 321 (or simply does not contain 321) otherwise.

Remark 4.18. Note that Definition 4.17 is different from the standard notion of pattern-containing or pattern-
avoiding permutations since here we require the one-line notation of w to contain the string 321 exactly.
Given a subset A = {j 1 < j 2 < · · · < j k } ⊆ {1, 2, . . . , n − 1} and corresponding Peterson-type fixed point w A , we call the permutation s j k s j k−1 · · · s j2 s j1 ∈ S n the Peterson case rolldown of w A . Note that the word (4.19) is in fact a reduced word decomposition of this permutation; we always use this choice of reduced word. The terminology is motivated by the fact that (4.19) is the (inverse of the) permutation given in [11,Definition 4.1]. (The fact that it is the inverse of the permutation used in [11] does not affect the theory very much, as is explained in [11,Proposition 5.16].) Lemma 4.19. Let n ≥ 4 and Hess(h) the 334-type Hessenberg variety in Fℓags(C n ). Let w be a Peterson-type fixed point and let A(w) = {j 1 < j 2 < · · · < j k } be its associated subset.
• Suppose w does not contain 321. Then roℓℓ(w) is the Peterson case rolldown of w A(w) .
In particular, if a Peterson-type fixed point w contains 321, then its rolldown roℓℓ(w) is Bruhat-greater, and has Bruhat length 1 greater, than the Peterson case rolldown of w.
Proof. If w contains a 321, then by Definition 2.6, the pairs (1, 3), (2,3) and (1,2) are all dimension pairs in w. Hence 3 appears precisely twice as a top part of a dimension pair and 2 appears precisely once. Thus by construction the dimension pair algorithm the permutation ω(x) begins with the word s 1 (s 2 s 1 ).
With respect to all other indices j ∈ A(w), the 334-type Hessenberg function is identical to the Peterson Hessenberg function and hence for each such j, the index j + 1 appears precisely once as a top part of a dimension pair of w and thus contributes precisely one s j to ω(x). Taking the inverse yields (4.20) as desired.
If w does not contain 321, then 3 appears at most once as the top part of a dimension pair in w, and again for all other indices the computations are identical to the Peterson case as above. Hence roℓℓ(w) is identical to the Peterson case rolldown. This completes the proof.
Next, we give an explicit description, along with a choice of reduced word decomposition, of the rolldowns corresponding to the non-Peterson-type fixed points. Lemma 4.20. Let w ∈ Hess(h) S 1 and suppose that w is not of Peterson type. Let A(w) = {j 1 = 1 < j 2 = 2 < j 3 < · · · < j k } for some k ≥ 2.
(1) If w is of 312-type, then the dimension pair algorithm associates to w the permutation (4.21) roℓℓ(w) = s j k s j k−1 · · · s j4 s j3 s 1 s 2 .
(2) If w is 231-type, then the dimension pair algorithm associates to w the permutation Proof. Suppose w is a 312-type fixed point so φ −1 λ (w −1 ) is a 312-type permissible filling. By definition of dimension pairs, 2 does not appear as the top part of any dimension pair (since it appears to the right of a 1). Also by definition, 3 appears as a top part of the two dimension pairs (1, 3) and (2, 3). The form of the 312-type permissible fillings described in Lemma 4.6 and the definition of A(w) imply that the other dimension pairs are precisely the pairs (j, j + 1) for j ∈ A(w) (for j = 1, 2), from which it follows that ω(x) = s 2 s 1 s j3 s j4 · · · s j k−1 s j k . Taking inverses yields (4.21). The proof of the second assertion is similar.
We conclude the section with a computation of the one-line notation of the rolldowns for different types; we leave proofs to the reader.

Bruhat order relations.
In this section we analyze the properties of the association w → A(w) with respect to comparisons in Bruhat order. The first two lemmas are straightforward and proofs left to the reader. We also observe that a Bruhat relation w < w ′ implies a containment relation of the associated subsets.
Lemma 4.25. Let w, w ′ ∈ Hess(h) S 1 and let A(w), A(w ′ ) be the respective associated subsets. Let s i be a simple transposition. Then: Proof. Bruhat order is independent of choice of reduced word decomposition for w. Therefore a simple transposition s i is less than w in Bruhat order if and only if s i appears in a (and hence any) reduced word decomposition of w. In particular, to prove the first claim it suffices to observe that by the definitions of A(w), the index i appears in A(w) precisely when s i appears in the choice of reduced word for w given above. A similar argument using the explicit reduced words given for roℓℓ(w) in Lemmas 4.19 and 4.20 proves the second claim. The last claim follows from the first two.
We have just seen that w ≤ w ′ implies A(w) ⊆ A(w ′ ). In the case of the Peterson variety Hess(h ′ ) these Bruhat relations are precisely encoded by the partial ordering given by containment of the A(w); specifically, by Lemma 4.23, w A ≤ w B if and only if A ⊆ B. In our 334-type Hessenberg case this is no longer true, although the sets A(w) do still encode the Bruhat data. The precise statements occupy the next several lemmas.
We take a moment to recall the tableau criterion for determining Bruhat order in the Weyl group S n (see e.g. [3]) which will be useful in the discussion below. For w ∈ S n , denote by D R (w) the descent set of w, namely,   Comparing corresponding entries, there are two violations of the tableau condition of the proposition (3 > 2) in the upper-left corner, so we conclude that w < v. Now we observe that some Bruhat relations never arise.
be the associated subset of w with its decomposition into maximal consecutive substrings. Suppose one of the following conditions hold: (1) w ′ is of Peterson type that does not contain 321 while w is not, (2) w ′ is 231-type while w is either of Peterson type that contains 321 or is 312-type. Then w < w ′ and roℓℓ(w) < w ′ .
Proof. If w ′ is of Peterson type that does not contain 321, then {1, 2} ⊆ A(w ′ ) by definition of the associated subsets. All other types (Peterson type that contains 321, or 312-type, or 231-type) have associated subsets containing {1, 2} by Lemma 4.13 and by definition of A(w). The claim (1) now follows from Lemma 4.25.
Next suppose w ′ is 231-type and w is of Peterson type that contains 321. Then the first two entries of the one-line notation of w must be both strictly greater than 1, and 2 ∈ D R (w). Similarly if w is a 312-type fixed point then a 2 ≥ 2. From (4.10) it follows that the first two entries in the one-line notation of w are also strictly greater than 1, and 2 ∈ D R (w). On the other hand, the one-line notation for a 231-type fixed point in (4.11) has a 1 in the second entry. By the tableau criterion, if w < w ′ then since 2 ∈ D R (w) in both cases under consideration, we must have that one of the first two entries of w is equal to 1, but we have just seen that is impossible. Hence w < w ′ . The assertion that roℓℓ(w) < w ′ follows by a similar argument using (4.23), (4.24), and (4.28).
For the next lemma and below, we say two fixed points are of the same type if both are Peterson-type, or both are 312-type, or both are 231-type. Lemma 4.28. Let w, w ′ ∈ Hess(h) S 1 . Suppose one of the following conditions hold: • w and w ′ are of the same type, or • w is of Peterson type and does not contain 321, and w ′ is either 312-type or 231-type, or • w is either 312-type or 231-type, and w ′ is of Peterson type.
Proof. Since the lemma above shows that w < w ′ implies A(w) ⊆ A(w ′ ), for all cases it suffices to show the reverse implication. First suppose w and w ′ are of the same type and A(w) ⊆ A(w ′ ). An examination of the reduced word decompositions of the 334-type fillings given in the above discussion and an argument similar to that in [11] implies w < w ′ . Now suppose w is of Peterson type and does not contain 321 and w ′ is either of 312 type or 231-type. Then since {1, 2} ⊆ A(w), either 1 ∈ A(w) or 2 ∈ A(w). From the explicit reduced word decompositions of 312 or 231-type fixed points chosen above it can be seen that w ′ is Bruhat-greater than both w A(w ′ )\{1} and w A(w ′ )\{2} . The claim now follows from Lemma 4.23. Finally suppose w is either 312-type or 231-type and w ′ is of Peterson type. Since A(w) ⊆ A(w ′ ) we know from Lemma 4.23 that w A(w) < w A(w ′ ) = w ′ . Lemma 4.24 shows that w < w A(w) so the result follows.
The next step is to show that Bruhat relations between certain Hessenberg fixed points are connected to lengths of initial maximal consecutive substrings in the associated subsets. We need some notation. Let A ⊆ {1, 2, . . . , n − 1}. Recall we denote by w A the Peterson-type fixed point associated to A. For the purposes of this discussion we let u A (respectively v A ) denote the 312-type (respectively 231-type) fixed point with associated subset A. Thus for A = [1, a] for some a with 2 ≤ a ≤ n − 1, we have u [1,a] = (a + 1 a · · · 3 1 2 a + 2 a + 3 · · · n) −1 = a a + 1 a − 1 a − 2 · · · 2 1 a + 2 a + 3 · · · n (4.27) and v [1,a] = (2 a + 1 a · · · 4 3 1 a + 2 a + 3 · · · n) −1 = a + 1 1 a a − 1 · · · 3 2 a + 2 a + 3 · · · n (4.28) in one-line notation. For general subsets with a 1 = 1 and a 2 ≥ 2, the definitions 312-type and 231-type fixed points imply that Proof. We begin by recalling two basic observations about Bruhat order in S n . Both follow straightforwardly from its definition in terms of reduced word decompositions. Suppose w, w ′ ∈ S n and assume that w and w ′ do not share any simple transpositions in their reduced word decompositions, i.e., s i < w implies s i < w ′ and vice versa. Then firstly, w · w ′ < w ′′ for w ′′ ∈ S n if and only if both w < w ′′ and w ′ < w ′′ . Secondly, w < w ′ · w ′′ if and only if w < w ′′ .
Recall that w A can be written as Moreover each factor appearing in the decomposition (4.31) (respectively (4.30) and (4.29)) for w A (respectively v A and u B ) has the property that it does not share any simple transpositions with any other factor appearing in the decomposition. Now suppose v A (respectively w A ) is Bruhat-less than u B . Then we know from Lemma 4.25 that A ⊆ B so it suffices to prove b 2 ≥ a 2 + 1. From Lemma 4.9 and the definition of the Peterson type fixed points we know that the one-line notation for v A (respectively w A ) has first a 2 + 1 entries a 2 + 1 1 a 2 a 2 − 1 · · · 3 2 (respectively a 2 + 1 a 2 a 2 − 1 · · · 3 2 1) while the one-line notation of u B has first b 2 + 1 entries given by In particular 1 ∈ D R (v A ) and also 1 ∈ D R (w A ). By the tableau criterion, this implies that the first entry of the one-line notation of v A and w A must be less than or equal to the first entry of that of u B . Hence a 2 + 1 ≤ b 2 as desired.
Conversely suppose A ⊆ B and b 2 ≥ a 2 + 1. Then an examination of the one-line notation of v where the last equality follows from Lemma 4.9. Since s 1 does not appear in any factor of w ′ the general fact above implies w ′ < u B . Finally since neither w [a1,a2] nor v [a1,a2] share any simple transpositions with w ′ the other general fact above yields v A < u B , w A < u B as desired.
be the respective associated subsets decomposed into maximal consecutive substrings. By Lemmas 3.5 and 4.4 it suffices to prove that if roℓℓ(w) ≤ w ′ , then w ≤ w ′ . So suppose roℓℓ(w) ≤ w ′ . By Lemma 4.25 this implies A(w) ⊆ A(w ′ ). By Lemma 4.28 we can conclude w ≤ w ′ if one of the following hold: • w and w ′ are of the same type, or • w is of Peterson type that does not contain 321, and w ′ is either 312-type or 231-type, or • w is either 312-type or 231-type, and w ′ is of Peterson type. Now suppose one of the following holds: • w ′ is of Peterson type that does not contain 321 and w is not, • w ′ is 231-type and w is of Peterson type that contains 321, or • w ′ is 231-type and w is 312-type.
In these cases, Lemma 4.27 implies that roℓℓ(w) < w ′ so there is nothing to prove. It remains to discuss the cases when: • w is of Peterson type that contains 321 and w ′ is 312-type, or • w is 231-type and w ′ is 312-type.
By Lemma 4.29 it suffices to show that a ′ 2 ≥ a 2 + 1. Suppose w is of Peterson type that contains 321. In particular a 2 ≥ 2. From (4.23) we know that 1 ∈ D R (roℓℓ(w)) and the first entry in the one-line notation of roℓℓ(w) is a 2 + 1. On the other hand (4.27) implies the one-line notation of w ′ begins with a ′ 2 . So if roℓℓ(w) < w ′ then the tableau criterion implies a ′ 2 ≥ a 2 + 1 as desired. Now suppose w is of 231-type. Again a 2 ≥ 2 and from (4.25) we know roℓℓ(w) has 1 ∈ D R (roℓℓ(w)) and a 2 + 1 as its first entry. By the same argument, a ′ 2 ≥ a 2 + 1 as desired. The result follows.

COMBINATORIAL FORMULAE FOR RESTRICTIONS TO FIXED POINTS OF 334-TYPE HESSENBERG SCHUBERT CLASSES
Our goal in this section is to give a combinatorial formula for p roℓℓ(w) (w) from which it follows as a corollary that it is nonzero. This proves Proposition 4.2 and hence Theorem 4.1. Although not strictly necessary for the proof of Proposition 4.2 we choose to prove the explicit formula (Proposition 5.9 below) since such a formula is a first step towards a derivation of a Monk formula for 334-type Hessenberg varieties and because it conveys a flavor of the combinatorics embedded in the GKM theory of Hessenberg varieties which are larger than the Peterson varieties in [11]. Many of our computations are analogues of those in [11,Section 5]. Our main tool is Billey's formula. We briefly recall some definitions and results (see also discussion in [11,Section 4]).  1 , b 2 , . . . , b ℓ(w) ) (corresponding to the word w = s b1 s b2 · · · s b ℓ(w) ) for w, define From the definition it follows that r(j, b) is an element of H * T (pt) ∼ = Sym(t * ) ∼ = C[t 1 , t 2 , . . . , t n ] of the form t ℓ − t k for some ℓ, k. These elements r(j, b) are the building blocks of Billey's formula [2,Theorem 4] which computes the restrictions σ v (w) of equivariant Schubert classes σ v at arbitrary permutations w in S n .

Theorem 5.2. ("Billey's formula", [2, Theorem 4])
Let w ∈ S n . Fix a reduced word decomposition w = s b1 s b2 · · · s b ℓ(w) and let b = (b 1 , b 2 , . . . , b ℓ(w) ) be the sequence of its indices. Let v ∈ S n . Then the restriction σ v (w) of the Schubert class σ v at the T -fixed point w is given by where the sum is taken over subwords s bj 1 s bj 2 · · · s bj ℓ(v) of b that are reduced words for v.
We record the following fact, used in the proof below, which follows straightforwardly from the Billey formula.

Fact 5.3.
Suppose v, w ∈ S n with v ≤ w in Bruhat order. Suppose there exists a decomposition w = w ′ · w ′′ for w ′ , w ′′ ∈ S n where v ≤ w ′ and, for all simple transpositions s i such that s i < v, we have s i ≤ w ′′ . Then Following terminology in [11], we refer to an individual summand of the expression in the right hand side of (5.2), corresponding to a single reduced subword v = s bj 1 s bj 2 · · · s bj ℓ(v) of w, as a summand in Billey's formula. In order to derive formulas for p v (w) where p v is a Hessenberg Schubert class, we use the linear projection π S 1 : t * → Lie(S 1 ) * dual to the inclusion of our circle subgroup S 1 into T given by (2.2). More specifically, since the diagram (4.6) commutes, we have We refer to the right hand side of the above equality as Billey's formula for p v (w). Recall π S 1 (t ℓ − t k+1 ) = (k + 1 − ℓ)t for a positive root t ℓ − t k+1 [11,Section 5]. We also use the following. We proceed to some preliminary computations. Let b = (b 1 , . . . , b ℓ(w) ) be a reduced word decomposition w = s b1 s b2 · · · s b ℓ(w) of w and let i be an index appearing in b, i.e. b ℓ = i for some 1 ≤ ℓ ≤ ℓ(w). Our first computation, Lemma 5.6, gives an expression for π S 1 (r(ℓ, b)) which shows in particular that the value of π S 1 (r(ℓ, b)) depends only on the value of the index b ℓ = i and not on its location ℓ in the word b. Note that if v = s i then the summands in Billey's formula for p v (w) = p si (w) are precisely equal to r(ℓ, b) for each ℓ such that b ℓ = i. Thus an equivalent formulation of the claim is that the summands in Billey's formula for p si (w) are all equal. This is analogous to a result in the Peterson case [11,Lemma 5.2] except that in our situation, the form of the formulas depend on the index i as well as on the type of the fixed point w in question.
(2) Suppose i ∈ A(w) and suppose one of the following conditions hold: • w is of Peterson type, or • w is 312-type, or • w is 231-type and i ∈ [a 1 , a 2 ]. Then each summand in Billey's formula for p si (w) is equal to (3) Suppose w is 231-type and i = 1. Then each summand in Billey's formula for p si (w) is equal to Proof. If i does not occur in A(w) then each summand is 0 by Billey's formula for σ si (w), since s i < w and thus never appears in the reduced word decomposition of w. For the next claim, the fact that each summand is equal to (i − T A(w) (i) + 1)t for the listed cases follows from examination of the chosen reduced word decompositions of w and an argument identical to that in [11]. Thus it remains to check the cases in which the summand differs from the case of Peterson varieties. First suppose w is 231-type and that i = 1. From the choice of explicit reduced word decomposition for such w given in (4.18) and Billey's formula, it follows that each summand in Billey's formula for σ s1 (w) is equal to r 2 (r 3 r 2 ) · · · (r a2−1 r a2−2 · · · r 3 r 2 )(r a2 r a2−1 · · · r 2 (t 1 − t 2 )) = r 2 (r 3 r 2 ) · · · (r a2−1 r a2−2 · · · r 3 r 2 )(t 1 − t a2+1 ) = t 1 − t a2+1 (5.4) since the reflection r j switches t j and t j+1 . Hence we have p s1 (w) = π S 1 (t 1 − t a2+1 ) = (a 2 + 1 − 1)t = a 2 t = H A(w) (1)t. Now suppose w is 231-type and i ∈ [2, a 2 ]. The factor in the reduced word decomposition (4.18) corresponding to [1 = a 1 , a 2 ] is equal to w [2,a2] · s 1 . By Fact 5.3, for i > 1 the presence of the extra s 1 does not affect the Billey computation, so each summand is equal to that for the Peterson type fixed point w [2,a2] and hence is equal to as desired.
Our next lemma concerns the summands in Billey's formula for p roℓℓ(w) (w) for w ∈ Hess(h) S 1 .
• Suppose w is 231-type. Then each summand for Billey's formula for p roℓℓ(w) (w) is equal to • Suppose w is of Peterson type that contains 321. Then each summand for Billey's formula for p roℓℓ(w) (w) is equal to   i∈A(w) Proof. Before considering the separate cases we make a general observation. By Lemma 5.6 and the discussion before Lemma 5.6 we know that the summands in Billey's formula for p si (w) for i ∈ A(w) are exactly the terms π S 1 (r(ℓ, b)) for ℓ such that b ℓ = i. Suppose in addition that w ∈ Hess(h) S 1 is such that roℓℓ(w) contains at most one simple transposition s i for each i ∈ {1, 2, . . . , n − 1}, i.e., roℓℓ(w) = s i1 s i2 · · · s i ℓ(roℓℓ(w)) is a reduced word for roℓℓ(w) where all i k are distinct for 1 ≤ k ≤ ℓ(roℓℓ(w)). This implies that any subword of a reduced word decomposition b of w which is a reduced word of roℓℓ(w) also must contain precisely one s i k for each 1 ≤ k ≤ ℓ(roℓℓ(w)). From Billey's formula (5.3) for p roℓℓ(w) (w) we know that a summand is of the form (5.5) π S 1 (r(j 1 , b))π S 1 (r(j 2 , b)) · · · π S 1 (r(j ℓ(v) , b)) where s bj 1 s bj 2 · · · s bj ℓ(roℓℓ(w)) is a reduced word of roℓℓ(w). Since {b j1 , . . . , b j ℓ(roℓℓ(w)) } = {i 1 , i 2 , . . . , i ℓ(roℓℓ(w)) } for each such summand the quantity (5.5) is equal to • Suppose w is 231-type. Then the number of summands in Billey's formula for p roℓℓ(w) (w) is 1.
Proof. We consider each case in turn. Suppose w is of Peterson type that contains 321. Let A(w) = [a 1 , a 2 ] ∪ [a 3 , a 4 ] ∪ · · · ∪ [a m−1 , a m ] be the decomposition of A(w) into maximal consecutive substrings. Recall a 2 ≥ 2 and a 1 = 1 in this case. The rolldown roℓℓ(w) is (s am s am−1 · · · s am−1 ) · · · (s a4 s a4−1 · · · s a3 ) · (s a2 s a2−1 · · · s 1 s 2 s 1 ) and w is w = w [a1,a2] w [a3,a4] · · · w [am−1,am] . Let ℓ > 1. As observed in the proof of Lemma 5.7 there is only one reduced word decomposition of the factor s a ℓ+1 s a ℓ+1 −1 · · · s a ℓ in roℓℓ(w). Moreover by examination it is evident that it appears only once in the standard reduced word decomposition of the corresponding w [a ℓ ,a ℓ+1 ] factor in w. Hence in order to count the number of ways roℓℓ(w) appears in w it suffices to count the number of subwords of the standard reduced word decomposition b of w [a1,a2] which are reduced subwords of s a2 s a2−1 · · · s 1 s 2 s 1 . We already saw in the proof of Lemma 5.7 that the reduced word s a2 s a2−1 · · · s 2 s 1 s 2 never appears in b. On the other hand since s 1 commutes with any s k with k ≥ 3, another reduced word decomposition of s a2 s a2−1 · · · s 1 s 2 s 1 is s 1 s a2 s a2−1 · · · s 2 s 1 . From examination of b it can be seen that the word s 1 s a2 s a2−1 · · · s 2 s 1 appears as a subword in the standard reduced word of w [a1,a2] precisely a 2 − 1 = H A(w) (1) − 1 times and that these are the only subwords of b which equal s a2 s a2−1 · · · s 1 s 2 s 1 . The claim follows.
Suppose w is of Peterson type that does not contain 321. Then the rolldown roℓℓ(w) is the Peterson case rolldown so the claim follows from explicit examination of the standard reduced word of w (alternatively from [11,Fact 4.5]).
Finally suppose w is 231-type. Then the rolldown roℓℓ(w) coincides with the Peterson case rolldown of w A(w) and the claim follows from examination of the reduced word decomposition (4.18).
The following is immediate from Lemmas 5.7 and 5.8. Proposition 5.9. Let w ∈ Hess(h) S 1 .
• Suppose w is of Peterson type that contains 321. Then • Suppose w is of Peterson type that does not contain 321. Then The proofs of the main results are now immediate.
Proof of Proposition 4.2. Let w ∈ Hess(h) S 1 . From the explicit formulas given in Proposition 5.9 it follows that p roℓℓ(w) (w) = 0 for all possible types of fixed points w.
Proof of Theorem 4.1. Since both (4.3) and (4.4) are satisfied for all w, w ′ ∈ Hess(h) S 1 by Propositions 4.3 and 4.2 respectively, the result follows.

OPEN QUESTIONS
This manuscript raises more questions than it answers. We close by mentioning some of them.  (Hess(N, h)).