Poset Pinball, the Dimension Pair Algorithm, and Type 𝐴 Regular Nilpotent Hessenberg Varieties

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 𝑋 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 𝑋. First we define the dimension pair algorithm, which yields 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. 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 𝐻^∗𝑆₁(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 ℂ coefficients), in certain cases of type 𝐴 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 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., [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 {𝜎_𝑤}_{𝑤∈𝑆𝑛} for 𝐻^∗𝑇(ℱℓ𝑎𝑔𝑠(ℂ^𝑛)) 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 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):Hess(𝑁,ℎ)∶=𝑉_•=0⊆𝑉₁⊆𝑉₂⊆⋯⊆𝑉_𝑛−1⊆𝑉_𝑛=ℂ^𝑛∣𝑁𝑉_𝑖⊆𝑉_ℎ(𝑖)∀𝑖=1,…,𝑛.(1.1) 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 paper: 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 [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 𝑆¹ 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 𝑆¹-fixed points Hess(𝑁,ℎ)^𝑆1 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 𝑆¹ into 𝑇 then induce a natural ring homomorphism𝐻^∗𝑇(ℱℓ𝑎𝑔𝑠(ℂ^𝑛))⟶𝐻^∗𝑆₁(Hess(𝑁,ℎ)).(1.2) 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^∗𝑆₁(Hess(𝑁,ℎ)) of the equivariant Schubert classes {𝜎_𝑤}_{𝑤∈𝑆𝑛} 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 ℱℓ𝑎𝑔𝑠(ℂ^𝑛)^𝑇≅𝑆_𝑛 (considered as a partially ordered set with respect to Bruhat order) and the subsetHess(𝑁,ℎ)^𝑆1⊆ℱℓ𝑎𝑔𝑠(ℂ^𝑛)^𝑇≅𝑆_𝑛(1.3) to determine a set of rolldowns in 𝑆_𝑛. 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 𝐻^∗𝑆₁(pt)-module basis of 𝐻^∗𝑆₁(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 [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 ℎ defined by ℎ(𝑖)=𝑖+1 for 1≤𝑖≤𝑛−1 and ℎ(𝑛)=𝑛. 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 𝐻^∗𝑆₁(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 [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 𝑆¹-fixed point 𝑤∈Hess(𝑁,ℎ)^𝑆1 a permutation in 𝑆_𝑛, which we call the rolldown of 𝑤 following terminology in [1] 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)𝑆1} 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 {𝑟𝑜ℓℓ(𝑤)}_{𝑤∈Hess(𝑁,ℎ)𝑆1} 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 𝑆¹-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 𝑆¹-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 𝐺𝐿(𝑛,ℂ)/𝐵 which is also identified withℱℓ𝑎𝑔𝑠(ℂ^𝑛)∶=𝑉_•={0}⊆𝑉₁⊆𝑉₂⊆⋯𝑉_𝑛−1⊆𝑉_𝑛=ℂ^𝑛∣dim_ℂ𝑉_𝑖=𝑖.(2.1) 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):Hess(𝑁,ℎ)∶=𝑉_•∈ℱℓ𝑎𝑔𝑠(ℂ^𝑛)∣𝑁𝑉_𝑖⊆𝑉_ℎ(𝑖)∀𝑖=1,…,𝑛⊆ℱℓ𝑎𝑔𝑠(ℂ^𝑛).(2.2) 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 𝑁 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 𝑁 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 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, that is, 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.
(ii) 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, 𝑟₁≤𝑟₂≤⋯≤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∶=𝜎𝑁𝜎⁻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=𝜎𝑁𝜎⁻¹ 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 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 [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 𝑁 be an 𝑛×𝑛 nilpotent matrix in Jordan canonical form and let 𝜎 denote a permutation matrix such that 𝑁_ℎ𝑓∶=𝜎𝑁𝜎⁻¹ is in highest form. It is known and straightforward to show that the following 𝑆¹ subgroup of 𝑈(𝑛,ℂ) preserves Hess(𝑁,ℎ) for 𝑁 as above and any Hessenberg function ℎ:𝑆¹=⎧⎪⎪⎪⎨⎪⎪⎪⎩⎡⎢⎢⎢⎢⎢⎢⎣𝑡^𝑛0𝑡0_𝑛−1⋯00000⋱00𝑡⎤⎥⎥⎥⎥⎥⎥⎦∣𝑡∈ℂ,‖𝑡‖=1⎫⎪⎪⎪⎬⎪⎪⎪⎭⊆𝑇_𝑛⊆𝑈(𝑛,ℂ).(2.3) Here 𝑇^𝑛 is the standard maximal torus of 𝑈(𝑛,ℂ) consisting of diagonal unitary matrices.

This implies that the conjugate circle subgroup 𝜎𝑆¹𝜎⁻¹ preserves Hess(𝑁_ℎ𝑓,ℎ). By abuse of notation we will denote both circle subgroups by 𝑆¹, since it is clear by context which is meant. The 𝑆¹-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)^𝑇𝑛 may be identified with the Weyl group 𝑊=𝑆_𝑛, and since Hess(𝑁,ℎ)^𝑆1 (resp., Hess(𝑁_ℎ𝑓,ℎ)^𝑆1) is a subset of ℱℓ𝑎𝑔𝑠(ℂ^𝑛)^𝑇𝑛, any Hessenberg fixed point may be thought of as a permutation 𝑤∈𝑆_𝑛.

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 𝑛×𝑛 complex matrices 𝑁 and Young diagrams (we use English notation for Young diagrams) 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 [14].

Definition 2.3. Let 𝜆 be a Young diagram with 𝑛 boxes. Let ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛} be a Hessenberg function. A filling of 𝜆 by the alphabet {1,2,…,𝑛} is an injective placing of the integers {1,2,…,𝑛} into the boxes of 𝜆. A filling of 𝜆 is called a (𝐡,𝜆)-permissible filling if for every horizontal adjacency

in the filling we have 𝑘≤ℎ(𝑗).

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

Example 2.5. Let 𝑛=5. Suppose 𝜆=(5) and ℎ=(3,3,4,5,5). Then

is a permissible filling, whereas

is not, since 4≰ℎ(1).
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.

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

For a dimension pair (𝑎,𝑏) of 𝑇, we will refer to 𝑏 as the top part of the dimension pair.

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

are (1,2), (1,3), and (1,4). 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 [14] and denote by 𝐷𝑃^𝑇 the set of dimension pairs of 𝑇. For each integer ℓ with 2≤ℓ≤𝑛, let 𝑥_ℓ∶=||(𝑎,ℓ)∣(𝑎,ℓ)∈𝐷𝑃^𝑇||(2.4) 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−1forall2≤ℓ≤𝑛. We call the integral vector 𝐱=(𝑥₂,𝑥₃,…,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 define𝑢_ℓ(𝐱)∶=𝑠^ℓ−11,𝑠_ℓ−2⋯𝑠_{ℓ−𝑥ℓ},if𝑥_ℓ>0,if𝑥_ℓ=0,(2.5) where 𝑠_𝑖 denotes the simple transposition (𝑖,𝑖+1) in 𝑆_𝑛 and 1 denotes the identity permutation. Now define the association𝐱⟼𝜔(𝐱)∶=𝑢₂(𝐱)𝑢₃(𝐱)⋯𝑢_𝑛(𝐱)∈𝑆_𝑛.(2.6) It is not difficult to see that (2.6) 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 (2.6) 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.

Fact 1. Suppose 𝐱=(𝑥₂,…,𝑥_𝑛), 𝐲=(𝑦₂,…,𝑦_𝑛)∈ℤ^𝑛−1≥0 are both lists of top parts. Suppose further that for all 2≤ℓ≤𝑛, we have 𝑥_ℓ≤𝑦_ℓ. Then 𝜔(𝐱)≤𝜔(𝐲) 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

, the set 𝐷𝑃^𝑇 of top parts of dimension pairs is {2,3,4}, yielding the integer vector 𝐱=(1,1,1,0). The associated permutation 𝜔(𝐱) is then 𝑠₁𝑠₂𝑠₃.

Example 2.9. Let 𝜆,ℎ be as in Example 2.5. The filling

is also permissible, with dimension pairs (1,2), (1,3), (1,4), (2,3). Hence 𝐱=(1,2,1,0) and the associated permutation 𝜔(𝐱) is 𝑠₁(𝑠₂𝑠₁)𝑠₃.

2.3. Bijection between Fixed Points and Permissible Fillings

For nilpotent Hessenberg varieties, the 𝑆¹-fixed points Hess(𝑁,ℎ)^𝑆1 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(2.7) has associated permutation 641523. It is easily seen that this is a bijective correspondence. Given a filling 𝑇 of 𝜆, we denote its associated permutation by 𝜙_𝜆(𝑇).

Remark 2.10. 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^𝜆−1(𝑤) by just 𝑤. For instance, the permissible filling of 𝜆=(5) in Example 2.9 has associated permutation 43215.
Now let 𝒫𝐹𝑖ℓℓ(𝜆,ℎ)(2.8) denote the set of (ℎ,𝜆)-permissible fillings of 𝜆. Recall that elements in Hess(𝑁,ℎ)^𝑆1 are viewed as permutations in 𝑆_𝑛 via the identification ℱℓ𝑎𝑔𝑠(ℂ^𝑛)^𝑇𝑛≅𝑆_𝑛. 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 𝑛 a positive integer. Let ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛} be a Hessenberg function and 𝜆 a Young diagram with 𝑛 boxes. Suppose 𝑁_ℎ𝑓 is a nilpotent operator in highest form as chosen in [16] (cf. Remark 2.2) with 𝜆_𝑁ℎ𝑓=𝜆. Let Hess(𝑁_ℎ𝑓,ℎ) denote the associated nilpotent Hessenberg variety. Then the map from the 𝑆¹-fixed points Hess(𝑁_ℎ𝑓,ℎ)^𝑆1 to the set of permissible fillings 𝒫𝐹𝑖ℓℓ(𝜆,ℎ)𝑤∈Hess(ℎ)^𝑆1⊆𝑆_𝑛⟼𝜙^𝜆−1𝑤⁻¹∈𝒫𝐹𝑖ℓℓ(𝜆,ℎ)(2.9) is well defined and is a bijection.

Remark 2.12. In the case when 𝑁 is the principal nilpotent 𝑛×𝑛 matrix, 𝜆 is the Young diagram with only one row. Thus the map (2.9) above simplifies to 𝑤↦𝑤⁻¹ where we abuse notation (cf. Remark 2.10) and denote 𝜙^𝜆−1(𝑤⁻¹) by 𝑤⁻¹.

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 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 [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 𝑁_ℎ𝑓 be a nilpotent 𝑛×𝑛 matrix in highest form chosen as in Remark 2.2 and let 𝜆∶=𝜆_𝑁ℎ𝑓. Let ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛} be a Hessenberg function and Hess(𝑁_ℎ𝑓,ℎ) the corresponding nilpotent Hessenberg variety.

The definition of the dimension pair algorithm is pure combinatorics. It produces for each Hessenberg fixed point 𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1 an element in 𝑆_𝑛. Following terminology of poset pinball, we denote this function by𝑟𝑜ℓℓ∶Hess𝑁_ℎ𝑓,ℎ^𝑆1⟶𝑆_𝑛.(3.1)

Definition 3.1 (the dimension pair algorithm). We define 𝑟𝑜ℓℓ∶Hess(𝑁_ℎ𝑓,ℎ)^𝑆1→𝑆_𝑛 as follows.(1)Let 𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1 and let 𝜙^𝜆−1(𝑤⁻¹) be its corresponding permissible filling as defined in (2.9).(2)Let 𝐷𝑃^{𝜙𝜆−1(𝑤−1)} be the set of dimension pairs in the permissible filling 𝜙^𝜆−1(𝑤⁻¹).(3)For each ℓ with 2≤ℓ≤𝑛, set 𝑥_ℓ∶=|||(𝑎,ℓ)∣(𝑎,ℓ)∈𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|||(3.2) as in (2.4) and define 𝐱∶=(𝑥₂,…,𝑥_𝑛).(4)Define 𝑟𝑜ℓℓ(𝑤)∶=(𝜔(𝐱))⁻¹ where 𝜔(𝐱) is the permutation associated to the integer vector 𝐱 defined in (2.6).

Example 3.2. Let 𝜆, ℎ be as in Example 2.5. The permutation 𝑤=43215∈𝑆_𝑛 is in Hess(𝑁_ℎ𝑓,ℎ)^𝑆1, as can be checked. The associated permissible filling is

. In Example 2.9 we saw that the associated permutation is 𝑠₁(𝑠₂𝑠₁)𝑠₃, so we conclude 𝑟𝑜ℓℓ(𝑤)=𝑠₃(𝑠₁𝑠₂)𝑠₁.

We next show that the rolldown function 𝑟𝑜ℓℓ∶Hess(ℎ)^𝑆1→𝑆_𝑛 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 𝑁_ℎ𝑓∶ℂ^𝑛→ℂ^𝑛 be a nilpotent matrix in highest form chosen as in Remark 2.2 and let 𝜆∶=𝜆_𝑁ℎ𝑓. Let ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛} be a Hessenberg function and let Hess(𝑁_ℎ𝑓,ℎ) denote the corresponding nilpotent Hessenberg variety. There is a paving by (complex) affine cells of Hess(𝑁_ℎ𝑓,ℎ) such that(i)the affine cells are in one-to-one correspondence with Hess(𝑁_ℎ𝑓,ℎ)^𝑆1, and(ii)the (complex) dimension of the affine cell 𝐶_𝑤 corresponding to a fixed point 𝑤∈Hess(𝑁,ℎ)^𝑆1 isdim_ℂ𝐶_𝑤=|||𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|||.(3.3)

In particular, Theorem 3.3 implies that the odd Betti numbers of Hess(𝑁_ℎ𝑓,ℎ) are 0, and the 2𝑘th even Betti number is precisely the number of fixed points 𝑤 in Hess(𝑁_ℎ𝑓,ℎ)^𝑆1 such that |𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|=𝑘. Given the regular nilpotent Hessenberg variety Hess(𝑁_ℎ𝑓,ℎ), denote by 𝑏_𝑘 its 2𝑘th Betti number, that is, 𝑏_𝑘∶=dim_ℂ𝐻^2𝑘Hess𝑁_ℎ𝑓,ℎ.(3.4) We may now formulate the conditions that guarantee that 𝑟𝑜ℓℓ∶Hess(𝑁_ℎ𝑓,ℎ)^𝑆1→𝑆_𝑛 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)𝑟𝑜ℓℓ∶Hess(𝑁_ℎ𝑓,ℎ)^𝑆1→𝑆_𝑛 is injective,(2)for every 𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1, we have 𝑟𝑜ℓℓ(𝑤)≤𝑤 in Bruhat order, and(3)for every 𝑘≥0, 𝑘∈ℤ, we have 𝑏_𝑘=|||𝑟𝑜ℓℓ(𝑤)∣𝑤∈Hess𝑁_ℎ𝑓,ℎ^𝑆1withℓ(𝑟𝑜ℓℓ(𝑤))=𝑘|||,(3.5) 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(𝑁_ℎ𝑓,ℎ)^𝑆1→𝑆_𝑛 is injective.

Proof. Since the association 𝐱=(𝑥₂,𝑥₃,…,𝑥_𝑛)↦𝜔(𝐱) given in (2.6) is a bijection, it suffices to show that the map which sends a Hessenberg fixed point 𝑤∈Hess(ℎ)^𝑆1 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) [14, Section 3.2]. The result follows.

Lemma 3.5. For every 𝑤∈Hess(ℎ)^𝑆1, one has 𝑟𝑜ℓℓ(𝑤)≤𝑤 in Bruhat order.

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)isaninversionin𝑢}(3.6) and let 𝐲∶=(𝑦₂,𝑦₃,…,𝑦_𝑛). Then the association (2.6) applied to the vector 𝐲 recovers the permutation 𝑢. By definition of 𝜙_𝜆 and the definition of dimension pairs, the set 𝐷𝑃^{𝜙𝜆−1(𝑤−1)} is always a subset of the set of inversions of the permutation 𝑤⁻¹. From Fact 1 it follows that the permutation 𝜔(𝐱) is Bruhat-less than 𝑤⁻¹ as desired.

Lemma 3.6. Let 𝑁_ℎ𝑓∶ℂ^𝑛→ℂ^𝑛 be a nilpotent matrix in highest form chosen as in Remark 2.2 and let 𝜆∶=𝜆_𝑁ℎ𝑓. Let ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛} be a Hessenberg function and Hess(𝑁_ℎ𝑓,ℎ) the associated nilpotent Hessenberg variety. For every 𝑘≥0, 𝑘∈ℤ, one has 𝑏_𝑘=|||roℓℓ(𝑤)∣𝑤∈Hess(ℎ)^𝑆1withℓ(roℓℓ(𝑤))=𝑘|||,(3.7) where ℓ(𝑟𝑜ℓℓ(𝑤)) denotes the Bruhat length of 𝑟𝑜ℓℓ(𝑤)∈𝑆_𝑛.

Proof. By construction, 𝑟𝑜ℓℓ(𝑤) has a reduced word decomposition consisting of precisely |𝐷𝑃^{𝜙𝜆−1(𝑤−1)}| simple transpositions. Hence its Bruhat length is |𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|. By Theorem 3.3, 𝑏_𝑘 is precisely the number of fixed points 𝑤 with |𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|=𝑘 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 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)^𝑆1⊆𝑆_𝑛, and target Betti numbers 𝑏_𝑘∶=dim_ℂ𝐻^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.

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 ℱℓ𝑎𝑔𝑠(ℂ^𝑛) with the standard 𝑇^𝑛-action and the GKM-compatible subspace is Hess(𝑁_ℎ𝑓,ℎ) with the 𝑆¹-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𝑝^{𝑟𝑜ℓℓ(𝑤)}∣𝑤∈Hess𝑁_ℎ𝑓,ℎ^𝑆1⊆𝐻^∗𝑆1Hess𝑁_ℎ𝑓,ℎ,(3.8) 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 map𝜋∶𝐻^∗𝑇_𝑛(ℱℓ𝑎𝑔𝑠(ℂ^𝑛))⟶𝐻^∗𝑆₁Hess𝑁_ℎ𝑓,ℎ(3.9) induced by the inclusion of groups 𝑆¹↪𝑇^𝑛 and the 𝑆¹-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 [1] we view 𝐻^∗𝑇_𝑛(ℱℓ𝑎𝑔𝑠(ℂ^𝑛)) and 𝐻^∗𝑆₁(Hess(𝑁_ℎ𝑓,ℎ)) as subrings of𝐻^∗𝑇_𝑛(ℱℓ𝑎𝑔𝑠(ℂ^𝑛))^𝑇𝑛≅_{𝑤∈𝑆𝑛}𝐻^∗𝑇_𝑛(pt)respectively𝐻^∗𝑆₁Hess𝑁_ℎ𝑓,ℎ^𝑆1≅_{𝑤∈Hess𝑁ℎ𝑓,ℎ𝑆1}𝐻^∗𝑆₁(pt).(3.10) 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)^𝑆1, respectively. If𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)≠0,𝑝_{𝑟𝑜ℓℓ(𝑤)}𝑤^=0if𝑤≰𝑤′(3.11) for all 𝑤, 𝑤′∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1, then the set {𝑝_{roℓℓ(𝑤)}∣𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1} in 𝐻^∗𝑆₁(Hess(𝑁_ℎ𝑓,ℎ)) is called poset-upper-triangular (with respect to the partial order on Hess(𝑁_ℎ𝑓,ℎ)^𝑆1⊆𝑆_𝑛 induced from Bruhat order) [1, 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 [1, Proposition  4.14] and the above discussion.

Proposition 3.9. Let Hess(𝑁_ℎ𝑓,ℎ) be either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Let roℓℓ∶Hess(𝑁_ℎ𝑓,ℎ)^𝑆1→𝑆_𝑛 be the dimension pair algorithm defined above. Suppose (3.11) holds for all 𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1. Then the Hessenberg Schubert classes {𝑝_{roℓℓ(𝑤)}∣𝑤∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1} form a 𝐻^∗𝑆₁(pt)-module basis for the 𝑆¹-equivariant cohomology ring 𝐻^∗𝑆₁(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 (3.11) for all 𝑤, 𝑤′∈Hess(𝑁_ℎ𝑓,ℎ)^𝑆1. 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], definedeg_{Hess(𝑁ℎ𝑓,ℎ)}(𝑤)∶=dim_ℂ𝐶_𝑤(3.12) 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 knowdeg_{Hess(𝑁ℎ𝑓,ℎ)}(𝑤)=|||𝐷𝑃^{𝜙𝜆−1(𝑤−1)}|||=ℓ(𝑟𝑜ℓℓ(𝑤)),(3.13) 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)^𝑆1→𝑆_𝑛 is also a matching in the sense of [1] with respect to deg_{Hess(𝑁ℎ𝑓,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)^𝑆1} 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 𝑆¹-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 𝒫𝐹𝑖ℓℓ(ℎ) instead of 𝒫𝐹𝑖ℓℓ(𝜆,ℎ).

We fix for this discussion the Hessenberg function given byℎ(1)=ℎ(2)=3,ℎ(𝑖)=𝑖+1for3≤𝑖≤𝑛−1,ℎ(𝑛)=𝑛.(4.1) The only difference between this function ℎ and the Hessenberg function for the Peterson variety studied in [13] 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 [13], although it is still necessary to introduce new ideas and terminology to handle the Hessenberg fixed points in Hess(ℎ)^𝑆1 which do not arise in the Peterson case.

The Hessenberg function ℎ in (4.1) is trivial if 𝑛=3 since in that case ℎ(1)=ℎ(2)=ℎ(3)=3 which implies that the corresponding Hessenberv variety Hess(ℎ) is equal to the full flag variety ℱℓ𝑎𝑔𝑠(ℂ³). 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.

Theorem 4.1. Let 𝑛≥4 and let Hess(ℎ) be the 334-type Hessenberg variety in ℱℓ𝑎𝑔𝑠(ℂ^𝑛). Let 𝑟𝑜ℓℓ∶Hess(ℎ)^𝑆1→𝑆_𝑛 be the dimension pair algorithm defined in Section 3. Then 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)≠0,𝑝_{𝑟𝑜ℓℓ(𝑤)}𝑤^=0if𝑤≰𝑤′(4.2) for all 𝑤, 𝑤′∈Hess(ℎ)^𝑆1. In particular the Hessenberg Schubert classes {𝑝_{𝑟𝑜ℓℓ(𝑤)}∣𝑤∈Hess(ℎ)^𝑆1} form a 𝐻^∗𝑆₁(pt)-module basis for the 𝑆¹-equivariant cohomology ring 𝐻^∗𝑆₁(Hess(ℎ)).

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 𝑛, ℎ, Hess(ℎ) and 𝑟𝑜ℓℓ be as above. Then, 𝑝_{roℓℓ(𝑤)}(𝑤)≠0(4.3) for all 𝑤∈Hess(ℎ)^𝑆1.

Proposition 4.3. Let 𝑛, ℎ, Hess(ℎ) and 𝑟𝑜ℓℓ be as above. Then, 𝑝_{𝑟𝑜ℓℓ(𝑤)}𝑤^=0if𝑤≰𝑤′(4.4) for all 𝑤, 𝑤′∈Hess(ℎ)^𝑆1.

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 𝑛, ℎ, Hess(ℎ) and 𝑟𝑜ℓℓ be as above. If for all 𝑤, 𝑤′∈Hess(ℎ)^𝑆1 one has 𝑟𝑜ℓℓ(𝑤)≤𝑤′⟺𝑤≤𝑤′(4.5) in Bruhat order, then the Hessenberg Schubert classes {𝑝_{𝑟𝑜ℓℓ(𝑤)}∣𝑤∈Hess(ℎ)^𝑆1} satisfy (4.4).

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 𝐻𝐻^∗𝑇_𝑛∗𝑆1(ℱℓ𝑎𝑔𝑠(ℂ^𝑛))𝐻^∗𝑇_𝑛(ℱℓ𝑎𝑔𝑠(ℂ^𝑛))^𝑇𝑛≅_𝑤∈𝑊𝐻^∗𝑇_𝑛(pt)(Hess(ℎ))𝐻^∗𝑆₁(Hess(ℎ))^𝑆1≅_{𝑤∈Hess(ℎ)𝑆1}𝐻^∗𝑆₁(pt)(4.6) commutes, it follows that if for all 𝑤, 𝑤′∈Hess(ℎ)^𝑆1, we have 𝑟𝑜ℓℓ(𝑤)≤𝑤′⟺𝑤≤𝑤′(4.7) 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)^𝑆1 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.

Lemma 4.5. Let 𝑛∈ℕ and let ℎ,ℎ′∶{1,2,…,𝑛}→{1,2,…,𝑛} be two Hessenberg functions. If ℎ(𝑖)≥ℎ′(𝑖) for all 𝑖, 1≤𝑖≤𝑛, then Hessℎ^⊆Hess(ℎ).(4.8) The inclusion Hess(ℎ′)↪Hess(ℎ) is 𝑆¹-equivariant and in particular Hess(ℎ′)^𝑆1⊆Hess(ℎ)^𝑆1 and 𝒫𝐹𝑖ℓℓ(ℎ′)⊆𝒫𝐹𝑖ℓℓ(ℎ).

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 Hessℎ^∶=𝑉_•∈ℱℓ𝑎𝑔𝑠(ℂ^𝑛)∣𝑁𝑉_𝑖⊆𝑉_ℎ^′_(𝑖),∀1≤𝑖≤𝑛,(4.9) where 𝑁 is the principal nilpotent operator. Since 𝑉_𝑖⊆𝑉_𝑖+1 for all 1≤𝑖≤𝑛−1 by definition of flags and 𝑉_𝑛=ℂ^𝑛 for all flags, if ℎ′(𝑖)≤ℎ(𝑖) for all 𝑖, then 𝑁𝑉_𝑖⊆𝑉_ℎ(𝑖) automatically implies 𝑁𝑉_𝑖⊆𝑉_ℎ(𝑖). We conclude Hess(ℎ′)⊆Hess(ℎ). The 𝑆¹-equivariance of the inclusion Hess(ℎ′)↪Hess(ℎ) follows from the definition of the 𝑆¹-action of (2.3).

Applying Lemma 4.5 to the Hessenberg functionℎ^(𝑖)=𝑖+1for1≤𝑖≤𝑛−1,ℎ^(𝑛)=𝑛(4.10) 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′)^𝑆1 also arise as fixed points in Hess(ℎ)^𝑆1. We refer to the elements of Hess(ℎ′)^𝑆1 (viewed as elements of Hess(ℎ)^𝑆1) as Peterson-type fixed points. It therefore remains to describe Hess(ℎ)^𝑆1⧵Hess(ℎ′)^𝑆1. It turns out to be convenient to do this by first describing 𝒫𝐹𝑖ℓℓ(ℎ)⧵𝒫𝐹𝑖ℓℓ(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 [13] that the 𝑆¹-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 𝒫𝐹𝑖ℓℓ(ℎ′) corresponding to Hess(ℎ′) are precisely those which are increasing sequences of staircases (cf. Remark 2.12). We now describe the permissible fillings 𝒫𝐹𝑖ℓℓ(ℎ) which are not Peterson-type fillings. We use the language of ℎ-tableau trees introduced by Mbirika; see [14, Section 3.1] for definitions. Recall from Remark 2.10 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∈𝒫𝐹𝑖ℓℓ(ℎ) be a permissible filling for Hess(ℎ) which is not of Peterson type, that is, 𝑤∈𝒫𝐹𝑖ℓℓ(ℎ)⧵𝒫𝐹𝑖ℓℓ(ℎ′). Then precisely one of the following hold.(i)The one-line notation of 𝑤 is of the form 𝑤^𝑤^312𝑤^,(4.11) 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. (ii)The one-line notation of 𝑤 is of the form 2𝑤^31𝑤^,(4.12) 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 𝒫𝐹𝑖ℓℓ(ℎ)⧵𝒫𝐹𝑖ℓℓ(ℎ′).

Proof of Lemma 4.6. For any Hessenberg function ℎ∶{1,2,…,𝑛}→{1,2,…,𝑛}, Mbirika shows in [14, Section 3.2] that the Level 𝑛 fillings in an ℎ-tableau tree are precisely the permissible fillings with respect to ℎ. For the Peterson Hessenberg function in (4.10) Mbirika’s corresponding ℎ-tableau tree has the property that for every 𝑘 with 1≤𝑘≤𝑛−1 and every vertex at Level 𝑘, there are precisely 2 edges going down from that vertex to a Level 𝑘+1 vertex (this is because the corresponding degree tuple 𝛽 [14, Definition  3.1.1] has 𝛽_𝑖=2 for all 1≤𝑖≤𝑛−1). In the case of the 334-type Hessenberg function, by definition the ℎ-tableau tree also has precisely 2 edges going down from every vertex at Level 𝑘 for all 𝑘≠2, 1≤𝑘≤𝑛−1. However, at Level 2, each vertex has not 2 but 3 edges pointing down to a vertex at Level 3.
From [14, Section 3] (cf. in particular [14, Definition 3.1.9]) it can be seen that for the case of the Peterson Hessenberg function, the corresponding ℎ-tableau tree at Level 2 has vertices •21• and 1•2•, whereas for the 334-type Hessenberg function, the Level 2 vertices have the form •2•1• and •1•2•. Here the bullets indicate the locations of the ℎ-permissible positions available for the placement of the next index 3, in the sense of [14, Section 3] (cf. in particular [14, Lemma  3.1.8]). 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 321•, 213• (coming from •2•1•) and the two vertices 1•32•, 12•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•31• and •312• 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.

Example 4.7. Suppose 𝑛=8. Then 𝑤=54312876 is an example of a 312-type permissible filling where 𝑤′=54 and 𝑤^=876. An example of a 231-type permissible filling is 𝑤=25431876 where 𝑤′=54 and 𝑤^=876. Neither of these are permissible with respect to the Peterson Hessenberg function ℎ′ since a 3 appears directly to the left of a 1. Nevertheless, both of these fillings are closely related to the Peterson-type permissible filling 𝑤=54321876; this relationship is closely analyzed and used below.

We now give explicit descriptions of the corresponding non-Peterson-type elements in Hess(ℎ)^𝑆1, obtained by taking inverses of the permissible fillings described in Lemma 4.6.

Definition 4.8. Let 𝑤∈Hess(ℎ)^𝑆1. We say 𝑤 is a 312-type (resp., 231-type) fixed point if its inverse 𝑤⁻¹ is a permissible filling of 312-type (resp., 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 (resp., 231-type) permissible filling. Let 𝑎₂ be the integer such that 𝑎₂+1 is the first entry (resp., second entry) in the one-line notation of 𝑤. Let 𝑤⁻¹ be the corresponding 312-type (resp., 231 type) fixed point. Then,(i)the one-line notation of 𝑤⁻¹ is the same as that of 𝑤 for all ℓth entries with ℓ>𝑎₂+1,(ii)if 𝑤 is 312-type, then the first 𝑎₂+1 entries of the one-line notation of 𝑤⁻¹ are 𝑎₂𝑎₂+1𝑎₂−1𝑎₂−2⋯21,(4.13)(iii)if 𝑤 is 231-type, then the first 𝑎₂+1 entries of the one-line notation of 𝑤⁻¹ are 𝑎₂+11𝑎₂𝑎₂−1⋯32.(4.14)

In the case of the Peterson variety, there is a convenient bijective correspondence between the set of 𝑆¹-fixed points of the Peterson variety and subsets 𝒜 of {1,2,…,𝑛−1} given as follows [13, Section 2.3]. Let 𝑤 be a Peterson-type fixed point. Then the corresponding subset is𝒜∶={𝑖∶1≤𝑖≤𝑛−1,𝑤(𝑖)=𝑤(𝑖+1)+1}⊆{1,2,…,𝑛−1}.(4.15) 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)^𝑆1. The associated subset of {1,2,…,𝑛}corresponding to 𝑤, notated 𝒜(𝑤), is defined as follows.(i)Suppose 𝑤 is of Peterson type. Then 𝒜(𝑤) is defined to be the set 𝒜 in (4.15).(ii)Suppose 𝑤 is 312-type. Consider the permutation 𝑤′∶=𝑤𝑠₁ (i.e., swap the 𝑎₂ and the 𝑎₂+1 in the one-line notation (4.13)). This is a fixed point of Peterson type. Define 𝒜(𝑤)∶=𝒜(𝑤′).(iii)Suppose 𝑤 is 231-type. Consider the permutation 𝑤′=𝑤𝑠₂𝑠₃⋯𝑠_𝑎2(4.16) (i.e., move the 1 to the right of the 2 in the one-line notation (4.14)). This is a fixed point of Peterson type.  Define 𝒜(𝑤)∶=𝒜(𝑤′).

Example 4.11. Suppose 𝑛=8.(i)Suppose 𝑤 is the Peterson-type fixed point 𝑤=54321876. Then 𝒜(𝑤)={1,2,3,4}∪{6,7}. This agrees with the association 𝑤↦𝒜(𝑤) used in [13].(ii)Suppose 𝑤 is the 312-type fixed point 𝑤=34217658 (corresponding to the 312-type permissible filling 43127658). Then 𝑤′=𝑤𝑠₁=43217658 and 𝒜(𝑤)∶=𝒜(𝑤′)={1,2,3}∪{5,6}.(iii)Suppose 𝑤 is the 231-type fixed point 𝑤=51432768 (corresponding to the 231-type permissible filling 25431768). Then 𝑤′=54321768 and 𝒜(𝑤)∶=𝒜(𝑤′)={1,2,3,4}∪{6}.

Remark 4.12. The three fixed points 𝑤=54321876, 𝑤=45321876, and 𝑤=51432876, which are, respectively, of Peterson type, 312 type, and 231-type, all have the same associated subset 𝒜(𝑤)={1,2,3,4}∪{6,7}.

It is useful to observe that the 312-type and 231-type fixed points have associated subsets that always contain 1 and 2.

Lemma 4.13. Let 𝑤 be a 334-type Hessenberg fixed point. Suppose further that 𝑤 is not of Peterson type. Then {1,2}⊆𝒜(𝑤).

Proof. From the explicit descriptions of the one-line notation of the 312 type (resp., 231-type) fixed points given above, we know that the initial segment 𝑎₂𝑎₂+1⋯21 (resp, 𝑎₂+11𝑎₂⋯32) in the one-line notation is such that 𝑎₂≥2. From Definition 4.10 it follows that the first decreasing staircase of the associated Peterson-type fixed point 𝑤𝑠₁ (resp., 𝑤𝑠₂𝑠₃⋯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 [13], but the notion is still useful for our arguments below.

4.2. 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)^𝑆1 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 nonnegative 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 by𝑠_𝑎𝑠_𝑎+1𝑠_𝑎𝑠_𝑎+2𝑠_𝑎+1𝑠_𝑎⋯𝑠_𝑎+𝑘𝑠_{𝑎+𝑘−1}⋯𝑠_𝑎+1𝑠_𝑎.(4.17) 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 [13, Section 2.3].) We denote a consecutive set of integers {𝑎,𝑎+1,…,𝑎+𝑘} for 𝑎 positive and 𝑘 a nonnegative integer by [𝑎,𝑎+𝑘]. We say that [𝑎,𝑎+𝑘] is a maximal consecutive substring of 𝒜 if [𝑎,𝑎+𝑘]⊆𝒜 and neither 𝑎−1 nor 𝑎+𝑘+1 are in 𝒜. It is straightforward that any subset 𝒜 of {1,2,…,𝑛−1} uniquely decomposes into a disjoint union of maximal consecutive substrings𝒜=𝑎₁,𝑎₂∪𝑎₃,𝑎₄∪⋯∪𝑎_𝑚−1,𝑎_𝑚.(4.18) For instance, for 𝒜={1,2,3,5,6,9,10,11}, the decomposition is 𝒜=[1,3]∪[5,6]∪[9,11]. For any [𝑎,𝑏], denote by 𝑤_[𝑎,𝑏] the full inversion in the subgroup 𝑆_{[𝑎,𝑏+1]}. Then it follows from Definition 4.10 (see also [13, Section 2.3]) that the Peterson-type fixed point associated to 𝒜, which we denote by 𝑤_𝒜, is the product𝑤_𝒜∶=𝑤_{[𝑎1,𝑎2]}𝑤_{[𝑎3,𝑎4]}𝑤_{[𝑎5,𝑎6]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(4.19) We fix a choice of reduced word decomposition of 𝑤_𝒜 given by taking the product of the standard reduced words (4.17) for each of the full inversions 𝑤_{[𝑎𝑗,𝑎𝑗+1]} appearing in (4.19). For the purposes of this manuscript we call this the standard reduced word decomposition of a Peterson-type fixed point 𝑤_𝒜.

Example 4.14. Let 𝑛=7 and let 𝑤=4321765 be a Peterson-type fixed point. Then the two decreasing staircases are 4321 and 765, the associated subset 𝒜(𝑤) is {1,2,3}∪{5,6} with maximal consecutive strings [1,3]∶={1,2,3} and [5,6]∶={5,6}. The standard reduced word decomposition of 𝑤 is 𝑤_{{1,2,3}∪{5,6}}=𝑤_[1,3]𝑤_[5,6]=𝑠₁𝑠₂𝑠₁𝑠₃𝑠₂𝑠₁𝑠₅𝑠₆𝑠₅.(4.20)

We now fix a reduced word decomposition of the non-Peterson-type fixed points.

Lemma 4.15. Let 𝑤∈Hess(ℎ)^𝑆1 be a fixed point which is not of Peterson type and let 𝒜(𝑤)=[𝑎₁,𝑎₂]∪[𝑎₃,𝑎₄]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] be the associated subset with its decomposition into maximal consecutive substrings.(i)If 𝑤 is 312-type, then a reduced word decomposition for 𝑤 is given by 𝑠₁𝑠₂𝑠₁⋯𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₃𝑠₂𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(4.21)(ii)If 𝑤 is 231-type, then a reduced word decomposition for 𝑤 is given by 𝑠₂𝑠₃𝑠₂⋯𝑠_𝑎2−1𝑠_𝑎2−2⋯𝑠₃𝑠₂𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚],}(4.22) where the 𝑤_{[𝑎ℓ,𝑎ℓ+1]} in the above expressions are assumed to be given the reduced word decomposition described in (4.17).

Proof. For the first assertion, observe that the explicit description of the one-line notation 312-type fixed points in (4.13) implies that 𝑤 has precisely 1 fewer inversion than 𝑤_𝒜(𝑤). An explicit computation shows that the given word (4.21) 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.

Example 4.16. Suppose 𝑛=7. Suppose 𝑤=3421765 is a 312-type fixed point. Then the reduced word decomposition of 𝑤 given in Lemma 4.15 is 𝑤=𝑠₁𝑠₂𝑠₁𝑠₃𝑠₂𝑠₅𝑠₆𝑠₅.(4.23) Similarly suppose 𝑤=4132765 is a 231-type fixed point. Then the reduced word decomposition of 𝑤 given in Lemma 4.15 is 𝑤=𝑠₂𝑠₃𝑠₂𝑠₁𝑠₅𝑠₆𝑠₅.(4.24)

Henceforth, we always use the reduced words given above.

Next we explicitly describe the rolldowns 𝑟𝑜ℓℓ(𝑤) associated to each 𝑤 in Hess(ℎ)^𝑆1 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 𝑤contains 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 𝑤does 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={𝑗₁<𝑗₂<⋯<𝑗_𝑘}⊆{1,2,…,𝑛−1} and corresponding Peterson-type fixed point 𝑤_𝒜, we call the permutation𝑠_𝑗𝑘𝑠_𝑗𝑘−1⋯𝑠_𝑗2𝑠_𝑗1∈𝑆_𝑛(4.25) the Peterson case rolldown of 𝑤_𝒜. Note that the word (4.25) 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 (4.25) is the (inverse of the) permutation given in [13, Definition  4.1]. (The fact that it is the inverse of the permutation used in [13] does not affect the theory very much, as is explained in [13, Proposition  5.16]).

Lemma 4.19. Let 𝑛≥4 and let Hess(ℎ) be the 334-type Hessenberg variety in ℱℓ𝑎𝑔𝑠(ℂ^𝑛). Let 𝑤 be a Peterson-type fixed point and let 𝒜(𝑤)={𝑗₁<𝑗₂<⋯<𝑗_𝑘} be its associated subset.(i)Suppose 𝑤 does not contain 321. Then 𝑟𝑜ℓℓ(𝑤) is the Peterson case rolldown of 𝑤_𝒜(𝑤).(ii)Suppose 𝑤 does contain 321, that is, 𝒜(𝑤)={𝑗₁<𝑗₂<⋯<𝑗_𝑘} for 𝑘≥2 and 𝑗₁=1 and 𝑗₂=2. Then 𝑟𝑜ℓℓ(𝑤) is𝑟𝑜ℓℓ(𝑤)=𝑠_𝑗𝑘𝑠_𝑗𝑘−1⋯𝑠_𝑗3𝑠₁𝑠₂𝑠₁.(4.26) 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 constructing the dimension pair algorithm the permutation 𝜔(𝐱) begins with the word 𝑠₁(𝑠₂𝑠₁). 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 (4.26) 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)^𝑆1 and suppose that 𝑤 is not of Peterson type. Let 𝒜(𝑤)={𝑗₁=1<𝑗₂=2<𝑗₃<⋯<𝑗_𝑘} for some 𝑘≥2.(1)If 𝑤 is of 312-type, then the dimension pair algorithm associates to 𝑤 the permutation 𝑟𝑜ℓℓ(𝑤)=𝑠_𝑗𝑘𝑠_𝑗𝑘−1⋯𝑠_𝑗4𝑠_𝑗3𝑠₁𝑠₂.(4.27)(2)If 𝑤 is 231-type, then the dimension pair algorithm associates to 𝑤 the permutation𝑟𝑜ℓℓ(𝑤)=𝑠_𝑗𝑘𝑠_𝑗𝑘−1⋯𝑠_𝑗4𝑠_𝑗3𝑠₂𝑠₁.(4.28)

Proof. Suppose 𝑤 is a 312-type fixed point so 𝜙^𝜆−1(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)=𝑠₂𝑠₁𝑠_𝑗3𝑠_𝑗4⋯𝑠_𝑗𝑘−1𝑠_𝑗𝑘. Taking inverses yields (4.27). The proof of the second assertion is similar.

Example 4.21. (i) Suppose 𝑤=54321876. This is of Peterson type. Then roℓℓ(𝑤)=(𝜔(𝐱))⁻¹=𝑠₇𝑠₆𝑠₄𝑠₃𝑠₁𝑠₂𝑠₁.(ii)Suppose 𝑤=45321876. This is 312-type. Then roℓℓ(𝑤)=(𝜔(𝐱))⁻¹=𝑠₇𝑠₆𝑠₄𝑠₃𝑠₁𝑠₂.(iii)Suppose 𝑤=51432876. This is 231-type. Then roℓℓ(𝑤)=(𝜔(𝐱))⁻¹=𝑠₇𝑠₆𝑠₄𝑠₃𝑠₂𝑠₁.

We conclude the section with a computation of the one-line notation of the rolldowns for different types; we leave proofs to the reader.

Lemma 4.22. Let 𝑤 be a 334-type Hessenberg fixed point and let 𝒜(𝑤)=[𝑎₁,𝑎₂]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] be its associated subset with its decomposition into maximal consecutive substrings. Suppose 𝑤 is of Peterson type that contains 321, 312-type, or of 231 type. Then 𝑎₁=1, 𝑎₂≥2 and the first 𝑎₂+1 entires of the one-line notation of 𝑟𝑜ℓℓ(𝑤) are 𝑎₂+12134⋯𝑎₂(4.29) for 𝑤 of Peterson type that contains 321, 2𝑎₂+1134⋯𝑎₂(4.30) for 𝑤312-type, and 𝑎₂+1123⋯𝑎₂(4.31) for 𝑤231-type.

4.3. 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 are left to the reader.

Lemma 4.23. Let 𝒜⊆{1,2,…,𝑛−1} and let 𝑤_𝒜 be the Peterson-type filling associated to 𝒜. Then 𝑤_𝒜 is maximal in the subgroup 𝑆_𝒜 of 𝑆_𝑛 generated by the simple transpositions {𝑠_𝑖}_𝑖∈𝒜. In particular, 𝑤_𝒜 is Bruhat-bigger than any permutation 𝑤∈𝑆_𝒜.

Lemma 4.24. Let 𝑤∈Hess(ℎ)^𝑆1. Suppose 𝑤 is not of Peterson type. Then 𝑤 is Bruhat-less than the Peterson type fixed point 𝑤_𝒜(𝑤) corresponding to 𝒜(𝑤).

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)^𝑆1 and let 𝒜(𝑤), 𝒜(𝑤′) be the respective associated subsets. Let 𝑠_𝑖 be a simple transposition. Then,(1)𝑠_𝑖<𝑤 if and only if 𝑖∈𝒜(𝑤),(2)𝑠_𝑖<𝑟𝑜ℓℓ(𝑤) if and only if 𝑖∈𝒜(𝑤),(3)if 𝑤≤𝑤′ or 𝑟𝑜ℓℓ(𝑤)≤𝑤^, then 𝒜(𝑤)⊆𝒜(𝑤′).

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., [17]) which will be useful in the discussion below. For 𝑤∈𝑆_𝑛, denote by 𝐷_𝑅(𝑤) the descent set of 𝑤, namely,𝐷_𝑅(𝑤)∶={𝑖∣𝑤(𝑖)>𝑤(𝑖+1),1≤𝑖≤𝑛−1}.(4.32) For example, for 𝑤=368475912 the descent set is 𝐷_𝑅(𝑤)={3,5,7}.

Theorem 4.26 (the tableau criterion [17, Theorem  2.6.3]). For 𝑤, 𝑣∈𝑆_𝑛, let 𝑤_𝑖,𝑘 be the 𝑖th element in the increasing rearrangement of 𝑤(1),𝑤(2),…,𝑤(𝑘), and similarly for 𝑣_𝑖,𝑘. Then 𝑤≤𝑣 in Bruhat order if and only if 𝑤_𝑖,𝑘≤𝑣_𝑖,𝑘,∀𝑘∈𝐷_𝑅(𝑤),1≤𝑖≤𝑘.(4.33)

For example, suppose 𝑤=368475912 and 𝑣=694287531. Since 𝐷_𝑅(𝑤)={3,5,7}, we examine the three increasing rearrangements of initial segments of 𝑤 and 𝑣 of lengths 3, 5, and 7, respectively, which we may organize into Young tableaux: (4.34)

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.

Lemma 4.27. Let 𝑤, 𝑤′∈Hess(ℎ)^𝑆1. Let 𝒜(𝑤)=[𝑎₁,𝑎₂]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] 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 𝑎₂≥2. From (4.13) 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 (4.14) 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 (4.29), (4.30), and (4.37).

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)^𝑆1. Suppose one of the following conditions hold:(i)𝑤 and 𝑤′ are of the same type, or(ii)𝑤 is of Peterson type and does not contain 321, and 𝑤′ is either 312-type or 231-type, or(iii)𝑤 is either 312-type or 231-type, and 𝑤′ is of Peterson type.Then, 𝑤<𝑤^iff𝒜(𝑤)⊆𝒜𝑤^.(4.35)

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 [13] 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 (resp., 𝑣_𝒜) denote the 312-type (resp., 231-type) fixed point with associated subset 𝒜. Thus for 𝒜=[1,𝑎] for some 𝑎 with 2≤𝑎≤𝑛−1, we have𝑢_[1,𝑎]=(𝑎+1𝑎⋯312𝑎+2𝑎+3⋯𝑛)⁻¹=𝑎𝑎+1𝑎−1𝑎−2⋯21𝑎+2𝑎+3⋯𝑛,(4.36)𝑣_[1,𝑎]=(2𝑎+1𝑎⋯431𝑎+2𝑎+3⋯𝑛)⁻¹=𝑎+11𝑎𝑎−1⋯32𝑎+2𝑎+3⋯𝑛(4.37) in one-line notation. For general subsets𝒜=𝑎₁,𝑎₂∪𝑎₃,𝑎₄∪⋯∪𝑎_𝑚−1,𝑎_𝑚,(4.38) with 𝑎₁=1 and 𝑎₂≥2, the definitions 312-type and 231-type fixed points imply that𝑢_𝒜=𝑢_{[𝑎1,𝑎2]}𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]},(4.39)𝑣_𝒜=𝑣_{[𝑎1,𝑎2]}𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(4.40)

Lemma 4.29. Let 𝒜, ℬ be subsets of {1,2,…,𝑛−1} and let 𝒜=𝑎₁,𝑎₂∪𝑎₃,𝑎₄∪⋯∪𝑎_𝑚−1,𝑎_𝑚,ℬ=𝑏₁,𝑏₂∪𝑏₃,𝑏₄∪⋯∪𝑏_𝑚−1,𝑏_𝑚(4.41) be the respective decompositions into maximal consecutive substrings. Assume both 𝒜 and ℬ contain {1,2}. Let 𝑤_𝒜 (resp., 𝑣_𝒜) be the Peterson-type (resp., 231-type) fixed point corresponding to 𝒜 and let 𝑢_ℬ be the 312-type fixed point corresponding to ℬ. Then 𝑤_𝒜<𝑢_ℬresp.,𝑣_𝒜<𝑢_ℬiff𝒜⊆ℬ,𝑏₂≥𝑎₂+1.(4.42)

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, that is, 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 𝑤_𝒜=𝑤_{[𝑎1,𝑎2]}⋅𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(4.43) Moreover, each factor appearing in the decomposition (4.43) (resp. (4.40) and (4.39)) for 𝑤_𝒜 (resp., 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 (resp., 𝑤_𝒜) is Bruhat-less than 𝑢_ℬ. Then we know from Lemma 4.25 that 𝒜⊆ℬ so it suffices to prove 𝑏₂≥𝑎₂+1. From Lemma 4.9 and the definition of the Peterson type fixed points we know that the one-line notation for 𝑣_𝒜 (resp., 𝑤_𝒜) has first 𝑎₂+1 entries 𝑎₂+11𝑎₂𝑎₂−1⋯32(4.44) (resp., 𝑎₂+1𝑎₂𝑎₂−1⋯321) while the one-line notation of 𝑢_ℬ has first 𝑏₂+1 entries given by 𝑏₂𝑏₂+1𝑏₂−1⋯321.(4.45) 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 𝑎₂+1≤𝑏₂ as desired.
Conversely suppose 𝒜⊆ℬ and 𝑏₂≥𝑎₂+1. Then an examination of the one-line notation of 𝑣_{[𝑎1,𝑎2]} (resp., 𝑤_{[𝑎1,𝑎2]}) compared to that of 𝑢_{[𝑏1,𝑏2]} and another application of the tableau criterion implies that 𝑣_{[𝑎1,𝑎2]}<𝑢_{[𝑏1,𝑏2]} and 𝑤_{[𝑎1,𝑎2]}<𝑢_{[𝑏1,𝑏2]}. In particular 𝑣_{[𝑎1,𝑎2]} and 𝑤_{[𝑎1,𝑎2]} are also Bruhat-less than 𝑢_ℬ. Moreover, since 𝒜⊆ℬ it follows that [𝑎₃,𝑎₄]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚]⊆ℬ so Lemma 4.23 implies 𝑤′∶=𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}<𝑤_ℬ=𝑢_ℬ⋅𝑠₁, where the last equality follows from Lemma 4.9. Since 𝑠₁ 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.

4.4. Proof of Proposition 4.3

We may now prove the upper-triangular vanishing property of 334-type Hessenberg Schubert classes.

Proof of Proposition 4.3. Let 𝑤,𝑤′∈Hess(ℎ)^𝑆1 and let 𝒜𝒜𝑤(𝑤_)=𝑎₁,𝑎₂∪𝑎₃,𝑎₄∪⋯∪𝑎_𝑚−1,𝑎_𝑚,=𝑎^1,𝑎^2∪𝑎^3,𝑎^4∪⋯∪𝑎^𝑟−1,𝑎^𝑟(4.46) 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 holds:(i)𝑤 and 𝑤′ are of the same type, or(ii)𝑤 is of Peterson type that does not contain 321, and 𝑤′ is either 312-type or 231-type, or(iii)𝑤 is either 312-type or 231-type, and 𝑤′ is of Peterson type.Now suppose one of the following holds:(i)𝑤′ is of Peterson type that does not contain 321 and 𝑤 is not,(ii)𝑤′ is 231-type and 𝑤 is of Peterson type that contains 321, or(iii)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(i)𝑤 is of Peterson type that contains 321 and 𝑤′ is 312-type, or(ii)𝑤 is 231-type and 𝑤′ is 312-type.By Lemma 4.29 it suffices to show that 𝑎^2≥𝑎₂+1. Suppose 𝑤 is of Peterson type that contains 321. In particular 𝑎₂≥2. From (4.29) we know that 1∈𝐷_𝑅(𝑟𝑜ℓℓ(𝑤)) and the first entry in the one-line notation of 𝑟𝑜ℓℓ(𝑤) is 𝑎₂+1. On the other hand (4.36) implies the one-line notation of 𝑤′ begins with 𝑎^2. So if 𝑟𝑜ℓℓ(𝑤)<𝑤^, then the tableau criterion implies 𝑎^2≥𝑎₂+1 as desired. Now suppose 𝑤 is of 231-type. Again 𝑎₂≥2 and from (4.31) we know 𝑟𝑜ℓℓ(𝑤) has 1∈𝐷_𝑅(𝑟𝑜ℓℓ(𝑤)) and 𝑎₂+1 as its first entry. By the same argument, 𝑎^2≥𝑎₂+1 as desired. The result follows.

5. 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 5.8 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 [13]. Many of our computations are analogues of those in [13, Section 5]. Our main tool is Billey’s formula. We briefly recall some definitions and results (see also discussion in [13, Section 4]).

Definition 5.1 (see [13, Definition  4.7]). Given a permutation 𝑤∈𝑆_𝑛, an index 𝑗∈{1,2,…,ℓ(𝑤)}, and a choice of reduced word decomposition 𝐛=(𝑏₁,𝑏₂,…,𝑏_ℓ(𝑤)) (corresponding to the word 𝑤=𝑠_𝑏1𝑠_𝑏2⋯𝑠_{𝑏ℓ(𝑤)}) for 𝑤, define 𝑟(𝑗,𝐛)∶=𝑠_𝑏1𝑠_𝑏2⋯𝑠_𝑏𝑗−1𝑡^𝑏_𝑗−𝑡_𝑏𝑗+1.(5.1)
From the definition it follows that 𝑟(𝑗,𝐛) is an element of 𝐻^∗𝑇(pt)≅Sym(𝑡^∗)≅ℂ[𝑡₁,𝑡₂,…,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 [18, 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, [18, Theorem  4]). Let 𝑤∈𝑆_𝑛. Fix a reduced word decomposition 𝑤=𝑠_𝑏1𝑠_𝑏2⋯𝑠_{𝑏ℓ(𝑤)} and let 𝐛=(𝑏₁,𝑏₂,…,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 𝜎_𝑣(𝑤)=𝑟𝑗₁,𝐛𝑟𝑗₂,𝐛⋯𝑟𝑗_ℓ(𝑣),𝐛,(5.2) where the sum is taken over subwords 𝑠_𝑏𝑗1𝑠_𝑏𝑗2⋯𝑠_{𝑏𝑗ℓ(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 2. 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 𝜎_𝑣(𝑤)=𝜎_𝑣(𝑤^).

Following terminology in [13], 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_𝑏𝑗1𝑠_𝑏𝑗2⋯𝑠_{𝑏𝑗ℓ(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_𝑆¹∶𝑡^∗→Lie(𝑆¹)^∗ dual to the inclusion of our circle subgroup 𝑆¹ into 𝑇 given by (2.3). More specifically, since the diagram (4.6) commutes, we have𝑝_𝑣(𝑤)=𝜋_𝑆¹𝑟𝑗₁,𝐛𝜋_𝑆¹𝑟𝑗₂,𝐛⋯𝜋_𝑆¹𝑟𝑗_ℓ(𝑣),𝐛.(5.3) We refer to the right hand side of the above equality as Billey’s formula for 𝑝_𝑣(𝑤). Recall 𝜋_𝑆¹(𝑡_ℓ−𝑡_𝑘+1)=(𝑘+1−ℓ)𝑡 for a positive root 𝑡_ℓ−𝑡_𝑘+1 [13, Section 5].

We also use the following.

Definition 5.3 (see [13, Definition 5.4]). Fix 𝒜⊆{1,2,…,𝑛−1}. Define ℋ_𝒜∶𝒜→𝒜 by ℋ_𝒜(𝑗)=themaximalelementinthemaximalconsecutivesubstringof𝒜containing𝑗.(5.4)

Definition 5.4 (see [13, Definition 5.5]). Fix 𝒜⊆{1,2,…,𝑛−1}. Define 𝒯_𝒜∶𝒜→𝒜 by 𝒯_𝒜(𝑗)=theminimalelementinthemaximalconsecutivesubstringof𝒜containing𝑗.(5.5)

We proceed to some preliminary computations. Let 𝐛=(𝑏₁,…,𝑏_ℓ(𝑤)) be a reduced word decomposition 𝑤=𝑠_𝑏1𝑠_𝑏2⋯𝑠_{𝑏ℓ(𝑤)} of 𝑤 and let 𝑖 be an index appearing in 𝐛, that is, 𝑏_ℓ=𝑖 for some 1≤ℓ≤ℓ(𝑤). Our first computation, Lemma 5.5, gives an expression for 𝜋_𝑆¹(𝑟(ℓ,𝐛)) which shows in particular that the value of 𝜋_𝑆¹(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 [13, 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.

Lemma 5.5. Let 𝑤∈Hess(ℎ)^𝑆1 and let 𝐛=(𝑏₁,…,𝑏_ℓ(𝑤)) be the reduced word decomposition of 𝑤 chosen in Section 4. Let 𝒜(𝑤)=[𝑎₁,𝑎₂]∪[𝑎₃,𝑎₄]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] be the associated subset of 𝑤 decomposed into maximal consecutive substrings. Let 𝑖∈{1,2,…,𝑛−1}.(1)If 𝑖∉𝒜(𝑤), then each summand in Billey’s formula for 𝑝_𝑠𝑖(𝑤) is 0. In particular, 𝑝_𝑠𝑖(𝑤)=0.(2)Suppose 𝑖∈𝒜(𝑤) and suppose one of the following conditions hold:(a)𝑤 is of Peterson type, or(b)𝑤 is 312-type, or(c)𝑤 is 231-type and 𝑖∉[𝑎₁,𝑎₂]. Then each summand in Billey’s formula for 𝑝_𝑠𝑖(𝑤) is equal to 𝑖−𝒯_𝒜(𝑤)(𝑖)+1𝑡.(5.6)(3)Suppose 𝑤 is 231-type and 𝑖=1. Then each summand in Billey’s formula for 𝑝_𝑠𝑖(𝑤) is equal to 𝑎₂𝑡=ℋ_𝒜(𝑤)(1)𝑡.(5.7)(4)Suppose 𝑤 if 231-type and 𝑖∈[2,𝑎₂]. Then each summand in Billey’s formula for 𝑝_𝑠𝑖(𝑤) is equal to 𝑖−𝒯_𝒜(𝑤)(𝑖)𝑡=(𝑖−1)𝑡.(5.8)

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 [13]. 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 (4.22) and Billey’s formula, it follows that each summand in Billey’s formula for 𝜎_𝑠1(𝑤) is equal to 𝑟₂𝑟₃𝑟₂⋯𝑟_𝑎2−1𝑟_𝑎2−2⋯𝑟₃𝑟₂𝑟_𝑎2𝑟_𝑎2−1⋯𝑟₂𝑡₁−𝑡₂==𝑟𝑡₂₁𝑟₃𝑟₂⋯𝑟_𝑎2−1𝑟_𝑎2−2⋯𝑟₃𝑟₂𝑡₁−𝑡_𝑎2+1−𝑡_𝑎2+1(5.9) since the reflection 𝑟_𝑗 switches 𝑡_𝑗 and 𝑡_𝑗+1. Hence we have 𝑝_𝑠1(𝑤)=𝜋_𝑆¹(𝑡₁−𝑡_𝑎2+1)=(𝑎₂+1−1)𝑡=𝑎₂𝑡=ℋ_𝒜(𝑤)(1)𝑡. Now suppose 𝑤 is 231-type and 𝑖∈[2,𝑎₂]. The factor in the reduced word decomposition (4.22) corresponding to [1=𝑎₁,𝑎₂] is equal to 𝑤_[2,𝑎2]⋅𝑠₁. By Fact 2, for 𝑖>1 the presence of the extra 𝑠₁ 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 𝑖−𝒯_𝒜(𝑤)(𝑖)=(𝑖−1)𝑡,(5.10) as desired.

Our next lemma concerns the summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) for 𝑤∈Hess(ℎ)^𝑆1.

Lemma 5.6. Let 𝑤∈Hess(ℎ)^𝑆1.(i)Suppose 𝑤 is 312-type or 𝑤 is of Peterson type that does not contain 321. Then each summand for Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) is equal to _{𝑖∈𝒜(𝑤)}𝑖−𝒯_𝒜(𝑤)(𝑖)+1⋅𝑡^|𝒜(𝑤)|.(5.11)(ii)Suppose 𝑤 is 231-type. Then each summand for Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) is equal to ℋ_𝒜(𝑤)(1)⋅⎛⎜⎜⎝^{ℋ𝒜(𝑤)(1)}_𝑖=2(𝑖−1)⎞⎟⎟⎠⋅⎛⎜⎜⎝_{𝑖∈𝒜(𝑤)⧵[𝒯𝒜(𝑤)(1),ℋ𝒜(𝑤)(1)]}𝑖−𝒯_𝒜(𝑤)(𝑖)+1⎞⎟⎟⎠𝑡^|𝒜(𝑤)|.(5.12)(iii)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)}𝑖−𝒯_𝒜(𝑤)(𝑖)+1⋅𝑡^{|𝒜(𝑤)|+1}.(5.13)

Proof. Before considering the separate cases we make a general observation. By Lemma 5.5 and the discussion before Lemma 5.5 we know that the summands in Billey’s formula for 𝑝_𝑠𝑖(𝑤) for 𝑖∈𝒜(𝑤) are exactly the terms 𝜋_𝑆¹(𝑟(ℓ,𝐛)) for ℓ such that 𝑏_ℓ=𝑖. Suppose in addition that 𝑤∈Hess(ℎ)^𝑆1 is such that 𝑟𝑜ℓℓ(𝑤) contains at most one simple transposition 𝑠_𝑖 for each 𝑖∈{1,2,…,𝑛−1}, that is, 𝑟𝑜ℓℓ(𝑤)=𝑠_𝑖1𝑠_𝑖2⋯𝑠_{𝑖ℓ(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_𝑖𝑘 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 𝜋_𝑆¹𝑟𝑗₁,𝐛𝜋_𝑆¹𝑟𝑗₂,𝐛⋯𝜋_𝑆¹𝑟𝑗_ℓ(𝑣),𝐛,(5.14) where 𝑠_𝑏𝑗1𝑠_𝑏𝑗2⋯𝑠_{𝑏𝑗ℓ(𝑟𝑜ℓℓ(𝑤))} is a reduced word of roℓℓ(𝑤). Since{𝑏_𝑗1,…,𝑏_{𝑗ℓ(𝑟𝑜ℓℓ(𝑤))}}={𝑖₁,𝑖₂,…,𝑖_{ℓ(𝑟𝑜ℓℓ(𝑤))}} for each such summand, the quantity (5.14) is equal to ^ℓ(𝑟^{𝑜ℓℓ(𝑤))}_𝑘=1𝑝_𝑠𝑖𝑘(𝑤).(5.15)
We now take cases. Suppose 𝑤 is not a Peterson-type that contains 321. Then from the explicit descriptions of 𝑟𝑜ℓℓ(𝑤) given in Section 4 it follows that 𝑟𝑜ℓℓ(𝑤) contains in its reduced word a single 𝑠_𝑖 for each 𝑖∈𝒜(𝑤). Thus we are in the situation described in the above paragraph and the claims follow from the computations given in Lemma 5.5.
Suppose 𝑤 is Peterson type and contains 321. Let 𝐛 be the standard reduced word decomposition (cf. (4.17) and (4.19)) of 𝑤. We claim that the only reduced word decompositions of 𝑟𝑜ℓℓ(𝑤) that occur as a subword of 𝐛 are those which contain two 𝑠₁’s, only one 𝑠₂, and precisely one 𝑠_𝑗ℓ for all other 𝑗_ℓ. Indeed let 𝒜(𝑤)=[𝑎₁,𝑎₂]∪[𝑎₃,𝑎₄]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] be the decomposition of 𝒜(𝑤) into maximal consecutive substrings. Recall 𝑎₂≥2 and 𝑎₁=1 in this case. The rolldown 𝑟𝑜ℓℓ(𝑤) is 𝑠_𝑎𝑚𝑠_𝑎𝑚−1⋯𝑠_𝑎𝑚−1⋯𝑠_𝑎4𝑠_𝑎4−1⋯𝑠_𝑎3⋅𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁(5.16) and 𝑤 is 𝑤=𝑤_{[𝑎1,𝑎2]}𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(5.17) Let ℓ>1. There is only one reduced word decomposition of the factor 𝑠_𝑎ℓ+1𝑠_{𝑎ℓ+1−1}⋯𝑠_𝑎ℓ in 𝑟𝑜ℓℓ(𝑤), so it remains to analyze the subwords of 𝑤_{[𝑎1,𝑎2]} which are reduced words of 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁. Let 𝐛 denote the standard reduced word of 𝑤_{[𝑎1,𝑎2]}. Note that another valid reduced word of 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁ is 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁𝑠₂. Since 𝑠₂ does not commute with 𝑠₁, the rightmost 𝑠₂ in the word 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁𝑠₂ must appear to the right of the 𝑠_𝑎2; in particular, there are two 𝑠₂’s to the right of the 𝑠_𝑎2 in this word. Since there is only one 𝑠₂ appearing to the right of the 𝑠_𝑎2 in 𝐛, we conclude that the reduced word of 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁ containing two copies of 𝑠₂ never appears as a subword of 𝐛. Hence the only subwords of 𝐛 contributing to summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) contain two 𝑠₁’s and one 𝑠₂, as claimed. Now since 𝑗₁=1 and 𝑗₂=2 and 𝑝_𝑠1(𝑤)=(1−𝒯_𝒜(𝑤)(1)+1)𝑡=(1−1+1)𝑡=𝑡 by Lemma 5.5, the claim follows.

We have just seen that all summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) are equal for all fixed points 𝑤∈Hess(ℎ)^𝑆1. In order to finish the computation we must now compute the number of summands which occur.

Lemma 5.7. Let 𝑤∈Hess(ℎ)^𝑆1.(i)Suppose 𝑤 is of Peterson type that contains 321. Then the number of summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) is ℋ_𝒜(𝑤)(1)−1.(ii)Suppose 𝑤 is of Peterson type that does not contain 321. Then the number of summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) is 1.(iii)Suppose 𝑤 is 312-type. Then the number of summands in Billey’s formula for 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤) is ℋ_𝒜(𝑤)(1)−1.(iv)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)=[𝑎₁,𝑎₂]∪[𝑎₃,𝑎₄]∪⋯∪[𝑎_𝑚−1,𝑎_𝑚] be the decomposition of 𝒜(𝑤) into maximal consecutive substrings. Recall 𝑎₂≥2 and 𝑎₁=1 in this case. The rolldown 𝑟𝑜ℓℓ(𝑤) is 𝑠_𝑎𝑚𝑠_𝑎𝑚−1⋯𝑠_𝑎𝑚−1⋯𝑠_𝑎4𝑠_𝑎4−1⋯𝑠_𝑎3⋅𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁(5.18) and 𝑤 is 𝑤=𝑤_{[𝑎1,𝑎2]}𝑤_{[𝑎3,𝑎4]}⋯𝑤_{[𝑎𝑚−1,𝑎𝑚]}.(5.19) Let ℓ>1. As observed in the proof of Lemma 5.6, there is only one reduced word decomposition of the factor 𝑠_𝑎ℓ+1𝑠_{𝑎ℓ+1−1}⋯𝑠_𝑎ℓ in 𝑟𝑜ℓℓ(𝑤). Moreover, by examination it is evident that it appears only once in the standard reduced word decomposition of the corresponding 𝑤_{[𝑎ℓ,𝑎ℓ+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_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁. We already saw in the proof of Lemma 5.6 that the reduced word 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁𝑠₂ never appears in 𝐛. On the other hand, since 𝑠₁ commutes with any 𝑠_𝑘 with 𝑘≥3, another reduced word decomposition of 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁ is 𝑠₁𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁. From examination of b it can be seen that the word 𝑠₁𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂𝑠₁ appears as a subword in the standard reduced word of 𝑤_{[𝑎1,𝑎2]} precisely 𝑎₂−1=ℋ_𝒜(𝑤)(1)−1 times and that these are the only subwords of 𝐛 which equal 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂𝑠₁. 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 [13, Fact  4.5]).
Suppose 𝑤 is 312-type. Then the rolldown 𝑟𝑜ℓℓ(𝑤) is of the form 𝑟𝑜ℓℓ(𝑤)=𝑠_𝑎𝑚𝑠_𝑎𝑚−1⋯𝑠_𝑎𝑚−1⋯𝑠_𝑎4𝑠_𝑎4−1⋯𝑠_𝑎3⋅𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂(5.20) from Lemma 4.20. By an argument similar to the case of Peterson type that contains 321 it suffices to analyze only the factors in both 𝑟𝑜ℓℓ(𝑤) and 𝑤 corresponding to the initial maximal consecutive substring [𝑎₁,𝑎₂]. As above we have 𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₁𝑠₂=𝑠₁𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂(5.21) and again it follows from examination of the standard reduced word of 𝑢_{[𝑎1,𝑎2]} that 𝑠₁𝑠_𝑎2𝑠_𝑎2−1⋯𝑠₂ appears precisely 𝑎₂−1=ℋ_𝒜(𝑤)(1)−1 times.
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 (4.22).

The following is immediate from Lemmas 5.6 and 5.7.

Proposition 5.8. Let 𝑤∈Hess(ℎ)^𝑆1.(i)Suppose 𝑤 is of Peterson type that contains 321. Then 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)=ℋ_𝒜(𝑤)(1)−1_{𝑖∈𝒜(𝑤)}𝑖−𝒯_𝒜(𝑤)(𝑖)+1⋅𝑡^{|𝒜(𝑤)|+1}.(5.22)(ii)Suppose 𝑤 is of Peterson type that does not contain 321. Then 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)=_{𝑖∈𝒜(𝑤)}𝑖−𝒯_𝒜(𝑤)(𝑖)+1𝑡^|𝒜(𝑤)|.(5.23)(iii)Suppose 𝑤 is of type 312. Then, 𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)=ℋ_𝒜(𝑤)(1)−1_{𝑖∈𝒜(𝑤)}𝑖−𝒯_𝒜(𝑤)(𝑖)+1⋅𝑡^|𝒜(𝑤)|.(5.24)(iv)Suppose 𝑤 is of type 231. Then,𝑝_{𝑟𝑜ℓℓ(𝑤)}(𝑤)=ℋ_𝒜(𝑤)(1)⋅⎛⎜⎜⎝^{ℋ𝒜(𝑤)(1)}_𝑖=2⋅⎛⎜⎜⎝(𝑖−1)_{𝑖∈𝒜(𝑤)⧵[𝒯𝒜(𝑤)(1),ℋ𝒜(𝑤)(1)]}⎞⎟⎟⎠×𝑖−𝒯_𝒜(𝑤)(𝑖)+1⎞⎟⎟⎠𝑡^|𝒜(𝑤)|.(5.25)

The proofs of the main results are now immediate.

Proof of Proposition 4.2. Let 𝑤∈Hess(ℎ)^𝑆1. From the explicit formulas given in Proposition 5.8 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)^𝑆1 by Propositions 4.3 and 4.2, respectively, the result follows.

6. Open Questions

This manuscript raises more questions than it answers. We close by mentioning some of them.

Question 1. For 𝑛≥4, Theorem 4.1 shows that for the case when 𝑁∶ℂ^𝑛→ℂ^𝑛 is the principal nilpotent operator and ℎ is the 334-type Hessenberg function, the dimension pair algorithm produces a set of Hessenberg Schubert classes {𝑝_{𝑟𝑜ℓℓ(𝑤)}}_{𝑤∈Hess(𝑁,ℎ)𝑆1} which are poset-upper-triangular and hence form a 𝐻^∗𝑆₁(pt)-module basis for 𝐻^∗𝑆₁(Hess(𝑁,ℎ)).(1)What are other examples of 𝑁 and ℎ such that the conclusion of Lemma 3.4 holds (cf. Remark 3.8)?(2)What are other examples of 𝑁 and ℎ for which the dimension pair algorithm produces a successful outcome of Betti poset pinball which is also poset-upper-triangular? Are there necessary and sufficient conditions on 𝑁 and ℎ that guarantee poset-upper-triangularity?(3)What are other examples of 𝑁 and ℎ for which the dimension pair algorithm produces a successful outcome of Betti poset pinball which corresponds to a linearly independent set of classes and hence a module basis? Are there necessary and sufficient conditions on 𝑁 and ℎ that guarantee this?

Question 2. In [13] the explicit module basis consisting of Peterson Schubert classes is used to derive a manifestly positive Monk formula in the 𝑆¹-equivariant cohomology of Peterson varieties. Preliminary investigation suggests that an analogous Monk formula for the 334-type Hessenberg varieties, using the module basis of Hessenberg Schubert classes derived in this manuscript, would be computationally much more complex. Thus, we may ask the following.(1)Does there exist a combinatorially elegant or computationally effective Monk formula for the 334-type Hessenberg varieties?(2)Can such a Monk formula be further generalized to a larger family of regular nilpotent Hessenberg varieties? For instance, can our techniques be generalized to give new insights to the equivariant Schubert calculus of the full flag variety ℱℓ𝑎𝑔𝑠(ℂ^𝑛) (which is an example of a regular nilpotent Hessenberg variety)?(3)In [12] the Monk formula for Peterson varieties is used to derive a Giambelli formula. Does there also exist a combinatorially elegant and/or computationally effective Giambelli formula for other cases of regular nilpotent Hessenberg varieties?

Acknowledgment

M. Harada is partially supported by an NSERC Discovery Grant, an NSERC University Faculty Award, and an Ontario Ministry of Research and Innovation Early Researcher Award.