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ISRN Geometry
Volumeย 2012ย (2012), Article IDย 254235, 34 pages
http://dx.doi.org/10.5402/2012/254235
Research Article

Poset Pinball, the Dimension Pair Algorithm, and Type ๐ด Regular Nilpotent Hessenberg Varieties

Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

Received 28 January 2012; Accepted 15 March 2012

Academic Editors: L. C.ย Jeffrey, A.ย Morozov, and E. H.ย Saidi

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

Abstract

We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace ๐‘‹ 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 ๐‘‹. First we define the dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type ๐ด regular nilpotent Hessenberg and any type ๐ด 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 ๐ปโˆ—๐‘†1(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 โ„‚ coefficients), in certain cases of type ๐ด 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, Definitionโ€‰โ€‰4.5]. In the context of this paper, the ambient GKM space is the flag variety โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›) equipped with the action of the diagonal subgroup ๐‘‡ of ๐‘ˆ(๐‘›,โ„‚), 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 {๐œŽ๐‘ค}๐‘คโˆˆ๐‘†๐‘› for ๐ปโˆ—๐‘‡(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)) 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 ๐‘โˆถโ„‚๐‘›โ†’โ„‚๐‘› be a nilpotent operator. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a function satisfying โ„Ž(๐‘–)โ‰ฅ๐‘– for all 1โ‰ค๐‘–โ‰ค๐‘› and โ„Ž(๐‘–+1)โ‰ฅโ„Ž(๐‘–) for all 1โ‰ค๐‘–<๐‘›. The associated Hessenberg variety Hess(๐‘,โ„Ž) is then defined as the following subvariety of โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›):Hess๎€ฝ๐‘‰(๐‘,โ„Ž)โˆถ=โ€ข=๎€ท0โŠ†๐‘‰1โŠ†๐‘‰2โŠ†โ‹ฏโŠ†๐‘‰๐‘›โˆ’1โŠ†๐‘‰๐‘›=โ„‚๐‘›๎€ธโˆฃ๐‘๐‘‰๐‘–โŠ†๐‘‰โ„Ž(๐‘–)๎€พ.โˆ€๐‘–=1,โ€ฆ,๐‘›(1.1) Since we deal exclusively with type ๐ด 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 ๐‘ is the principal nilpotent operator (in this case Hess(๐‘,โ„Ž) is called a regular nilpotent Hessenberg variety) and when โ„Ž is the identity function โ„Ž(๐‘–)=๐‘– for all 1โ‰ค๐‘–โ‰ค๐‘› (in this case Hess(๐‘,โ„Ž) is called a nilpotent Springer variety and is sometimes denoted ๐’ฎ๐‘). Hessenberg varieties arise in many areas of mathematics, including geometric representation theory [3โ€“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 ๐‘†1 subgroup of the unitary diagonal matrices ๐‘‡ which acts on Hess(๐‘,โ„Ž) (defined precisely in Section 2). The group ๐‘‡, the maximal torus of ๐‘ˆ(๐‘›,โ„‚), acts on โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›) in the standard fashion. It turns out that the ๐‘†1-fixed points Hess(๐‘,โ„Ž)๐‘†1 are a subset of the ๐‘‡-fixed points โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡โ‰…๐‘†๐‘›. Moreover, the inclusion of Hess(๐‘,โ„Ž) into โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›) and the inclusion of groups ๐‘†1 into ๐‘‡ then induce a natural ring homomorphism๐ปโˆ—๐‘‡(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›))โŸถ๐ปโˆ—๐‘†1(Hess(๐‘,โ„Ž)).(1.2) As mentioned above, it is well known in Schubert calculus that the equivariant Schubert classes {๐œŽ๐‘ค}๐‘คโˆˆ๐‘†๐‘› are a computationally convenient ๐ปโˆ—๐‘‡(pt)-module basis for ๐ปโˆ—๐‘‡(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)). We refer to the images in ๐ปโˆ—๐‘†1(Hess(๐‘,โ„Ž)) of the equivariant Schubert classes {๐œŽ๐‘ค}๐‘คโˆˆ๐‘†๐‘› 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 โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡โ‰…๐‘†๐‘› (considered as a partially ordered set with respect to Bruhat order) and the subsetHess(๐‘,โ„Ž)๐‘†1โŠ†โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡โ‰…๐‘†๐‘›(1.3) to determine a set of rolldowns in ๐‘†๐‘›. 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 ๐ปโˆ—๐‘†1(pt)-module basis of ๐ปโˆ—๐‘†1(Hess(๐‘,โ„Ž)). 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 goalโ€”that is, of constructing a module basis via poset pinball techniquesโ€”in the special case of the Peterson variety, which is the regular nilpotent Hessenberg variety with Hessenberg function โ„Ž defined by โ„Ž(๐‘–)=๐‘–+1 for 1โ‰ค๐‘–โ‰ค๐‘›โˆ’1 and โ„Ž(๐‘›)=๐‘›. Exploiting this explicit module basis, in [13, Theoremโ€‰โ€‰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 ๐ปโˆ—๐‘†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 [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 ๐‘†1-fixed point ๐‘คโˆˆHess(๐‘,โ„Ž)๐‘†1 a permutation in ๐‘†๐‘›, which we call the rolldown of ๐‘ค following terminology in [1] and denoted ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆˆ๐‘†๐‘›. In the special cases of regular nilpotent Hessenberg varieties and nilpotent Springer varieties, we show that the set {๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)}๐‘คโˆˆHess(๐‘,โ„Ž)๐‘†1 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, Sectionโ€‰โ€‰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 {๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)}๐‘คโˆˆHess(๐‘,โ„Ž)๐‘†1 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 ๐‘†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 [15]. Although we do not know whether the dimension pair algorithm always succeeds in producing module bases for the ๐‘†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 [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 ๐บ๐ฟ(๐‘›,โ„‚)/๐ต which is also identified withโ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›๎€ฝ๐‘‰)โˆถ=โ€ข=๎€ท{0}โŠ†๐‘‰1โŠ†๐‘‰2โŠ†โ‹ฏ๐‘‰๐‘›โˆ’1โŠ†๐‘‰๐‘›=โ„‚๐‘›๎€ธโˆฃdimโ„‚๎€ท๐‘‰๐‘–๎€ธ๎€พ.=๐‘–(2.1) A Hessenberg function is a function โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} satisfying โ„Ž(๐‘–)โ‰ฅ๐‘– for all 1โ‰ค๐‘–โ‰ค๐‘› and โ„Ž(๐‘–+1)โ‰ฅโ„Ž(๐‘–) for all 1โ‰ค๐‘–<๐‘›. We frequently denote a Hessenberg function by listing its values in sequence, โ„Ž=(โ„Ž(1),โ„Ž(2),โ€ฆ,โ„Ž(๐‘›)=๐‘›). Let ๐‘โˆถโ„‚๐‘›โ†’โ„‚๐‘› be a linear operator. The Hessenberg variety Hess(๐‘,โ„Ž) is defined as the following subvariety of โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›):Hess๎€ฝ๐‘‰(๐‘,โ„Ž)โˆถ=โ€ขโˆˆโ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)โˆฃ๐‘๐‘‰๐‘–โŠ†๐‘‰โ„Ž(๐‘–)๎€พโˆ€๐‘–=1,โ€ฆ,๐‘›โŠ†โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›).(2.2) If ๐‘ is nilpotent, we say Hess(๐‘,โ„Ž) is a nilpotent Hessenberg variety, and if ๐‘ is the principal nilpotent operator (i.e., has one Jordan block with eigenvalue 0), then Hess(๐‘,โ„Ž) is called a regular nilpotent Hessenberg variety. If ๐‘ is nilpotent and โ„Ž is the identity function โ„Ž(๐‘–)=๐‘– for all 1โ‰ค๐‘–โ‰ค๐‘›, then Hess(๐‘,โ„Ž) is called a nilpotent Springer variety and often denoted ๐’ฎ๐‘. In this manuscript we study in some detail the regular nilpotent case, and as such sometimes notate Hess(๐‘,โ„Ž) as Hess(โ„Ž) when ๐‘ is understood to be the standard principal nilpotent operator.

Suppose ๐‘ 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 ๐‘ which is in highest form [16, Definitionโ€‰โ€‰4.2]. We recall the following.

Definition 2.1 (see [16, Definitionsโ€‰4.1 andโ€‰4.2]). (i) Let ๐‘‹ be any ๐‘šร—๐‘› matrix. We call the entry ๐‘‹๐‘–๐‘˜ a pivot of ๐‘‹ if ๐‘‹๐‘–๐‘˜ is nonzero and if all entries below and to its left vanish, that is, ๐‘‹๐‘–๐‘—=0 if ๐‘—<๐‘˜ and ๐‘‹๐‘—๐‘˜=0 if ๐‘—>๐‘–. Moreover, given ๐‘–, define ๐‘Ÿ๐‘– to be the row of ๐‘‹๐‘Ÿ๐‘–,๐‘– if the entry is a pivot, and 0 otherwise.
(ii) Let ๐‘ be an upper-triangular nilpotent ๐‘›ร—๐‘› matrix. Then we say ๐‘ is in highest form if its pivots form a nondecreasing sequence, namely, ๐‘Ÿ1โ‰ค๐‘Ÿ2โ‰คโ‹ฏโ‰ค๐‘Ÿ๐‘›.

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 ๐‘ is the principal nilpotent matrix, then ๐‘ is already in highest form, and secondly that any nilpotent matrix can be conjugated by an appropriate ๐‘›ร—๐‘› permutation matrix ๐œŽ so that ๐‘โ„Ž๐‘“โˆถ=๐œŽ๐‘๐œŽโˆ’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 ๐‘โ„Ž๐‘“=๐œŽ๐‘๐œŽโˆ’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 ๐‘ is principal nilpotent, we take ๐‘โ„Ž๐‘“=๐‘ since ๐‘ 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 ๐‘ be an ๐‘›ร—๐‘› nilpotent matrix in Jordan canonical form and let ๐œŽ denote a permutation matrix such that ๐‘โ„Ž๐‘“โˆถ=๐œŽ๐‘๐œŽโˆ’1 is in highest form. It is known and straightforward to show that the following ๐‘†1 subgroup of ๐‘ˆ(๐‘›,โ„‚) preserves Hess(๐‘,โ„Ž) for ๐‘ as above and any Hessenberg function โ„Ž:๐‘†1=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉโŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐‘ก๐‘›0โ‹ฏ00๐‘ก๐‘›โˆ’10โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆโŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญ00โ‹ฑ000๐‘กโˆฃ๐‘กโˆˆโ„‚,โ€–๐‘กโ€–=1โŠ†๐‘‡๐‘›โŠ†๐‘ˆ(๐‘›,โ„‚).(2.3) Here ๐‘‡๐‘› is the standard maximal torus of ๐‘ˆ(๐‘›,โ„‚) consisting of diagonal unitary matrices.

This implies that the conjugate circle subgroup ๐œŽ๐‘†1๐œŽโˆ’1 preserves Hess(๐‘โ„Ž๐‘“,โ„Ž). By abuse of notation we will denote both circle subgroups by ๐‘†1, since it is clear by context which is meant. The ๐‘†1-fixed points of Hess(๐‘,โ„Ž) and Hess(๐‘โ„Ž๐‘“,โ„Ž) are isolated and are a subset of the ๐‘‡๐‘›-fixed points of โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). Since the set of ๐‘‡๐‘›-fixed points โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡๐‘› may be identified with the Weyl group ๐‘Š=๐‘†๐‘›, and since Hess(๐‘,โ„Ž)๐‘†1 (resp., Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1) is a subset of โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡๐‘›, any Hessenberg fixed point may be thought of as a permutation ๐‘คโˆˆ๐‘†๐‘›.

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 ๐‘›ร—๐‘› complex matrices ๐‘ and Young diagrams (we use English notation for Young diagrams) with ๐‘› boxes, given by associating to ๐‘ the Young diagram ๐œ† with row lengths the sizes of the Jordan blocks of ๐‘ listed in weakly decreasing order. We will use this bijection to often treat such ๐‘ and ๐œ† as the same data; we sometimes denote by ๐œ†๐‘ the Young diagram given as above corresponding to a nilpotent ๐‘.

For more details on the following see [14].

Definition 2.3. Let ๐œ† be a Young diagram with ๐‘› boxes. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function. A filling of ๐œ† by the alphabet {1,2,โ€ฆ,๐‘›} is an injective placing of the integers {1,2,โ€ฆ,๐‘›} into the boxes of ๐œ†. A filling of ๐œ† is called a (๐ก,๐œ†)-permissible filling if for every horizontal adjacency

254235.fig.001
in the filling we have ๐‘˜โ‰คโ„Ž(๐‘—).

Remark 2.4. In this manuscript the ๐œ† and โ„Ž will frequently be understood by context. When there is no danger of confusion we simply refer to permissible fillings.

Example 2.5. Let ๐‘›=5. Suppose ๐œ†=(5) and โ„Ž=(3,3,4,5,5). Then

254235.fig.002
is a permissible filling, whereas
254235.fig.003
is not, since 4โ‰ฐโ„Ž(1).
We denote a permissible filling of ๐œ† by ๐‘‡, in analogy with standard notation for Young tableaux. Next we focus attention on certain pairs of entries in a permissible filling ๐‘‡.

Definition 2.6. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function and ๐œ† a Young diagram with ๐‘› boxes. A pair (๐‘Ž,๐‘) is a dimension pair of an (โ„Ž,๐œ†)-permissible filling ๐‘‡ of ๐œ† if the following conditions hold:(1)๐‘>๐‘Ž, (2)๐‘ is either(i)below ๐‘Ž in the same column of ๐‘Ž, or(ii)anywhere in a column strictly to the left of the column of ๐‘Ž, and(3)if there exists a box with filling ๐‘ directly adjacent to the right of ๐‘Ž, then ๐‘โ‰คโ„Ž(๐‘).

For a dimension pair (๐‘Ž,๐‘) of ๐‘‡, we will refer to ๐‘ as the top part of the dimension pair.

Example 2.7. Let ๐œ†, โ„Ž be as in Example 2.5. The dimension pairs in the permissible filling

254235.fig.004
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 4โ‰ฐโ„Ž(1).

Given a permissible filling ๐‘‡ of ๐œ†, we follow [14] and denote by ๐ท๐‘ƒ๐‘‡ the set of dimension pairs of ๐‘‡. For each integer โ„“ with 2โ‰คโ„“โ‰ค๐‘›, let ๐‘ฅโ„“||๎€ฝโˆถ=(๐‘Ž,โ„“)โˆฃ(๐‘Ž,โ„“)โˆˆ๐ท๐‘ƒ๐‘‡๎€พ||(2.4) so ๐‘ฅโ„“ is the number of times โ„“ occurs as a top part in the set of dimension pairs of ๐‘‡. From the definitions it follows that 0โ‰ค๐‘ฅโ„“โ‰คโ„“โˆ’1forall2โ‰คโ„“โ‰ค๐‘›. We call the integral vector ๐ฑ=(๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›) the list of top parts of ๐‘‡.

To each such ๐ฑ we associate a permutation in ๐‘†๐‘› as follows. As a preliminary step, for each โ„“ with 2โ‰คโ„“โ‰ค๐‘› define๐‘ขโ„“(๎‚ป๐‘ ๐ฑ)โˆถ=โ„“โˆ’1๐‘ โ„“โˆ’2โ‹ฏ๐‘ โ„“โˆ’๐‘ฅโ„“,if๐‘ฅโ„“>0,1,if๐‘ฅโ„“=0,(2.5) where ๐‘ ๐‘– denotes the simple transposition (๐‘–,๐‘–+1) in ๐‘†๐‘› and 1 denotes the identity permutation. Now define the association๐ฑโŸผ๐œ”(๐ฑ)โˆถ=๐‘ข2(๐ฑ)๐‘ข3(๐ฑ)โ‹ฏ๐‘ข๐‘›(๐ฑ)โˆˆ๐‘†๐‘›.(2.6) It is not difficult to see that (2.6) is a bijection between the set of integral vectors ๐ฑโˆˆโ„ค๐‘›โˆ’1 satisfying 0โ‰ค๐‘ฅโ„“โ‰คโ„“โˆ’1 for all 2โ‰คโ„“โ‰ค๐‘›โˆ’1 and the group ๐‘†๐‘›. In fact the word given by (2.6) is a reduced word decomposition of ๐œ”(๐ฑ) and the ๐‘ฅโ„“ count the number of inversions in ๐œ”(๐ฑ) with โ„“ as the higher integer. The following simple fact will be used later.

Fact 1. Suppose ๐ฑ=(๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›), ๐ฒ=(๐‘ฆ2,โ€ฆ,๐‘ฆ๐‘›)โˆˆโ„ค๐‘›โˆ’1โ‰ฅ0 are both lists of top parts. Suppose further that for all 2โ‰คโ„“โ‰ค๐‘›, we have ๐‘ฅโ„“โ‰ค๐‘ฆโ„“. Then ๐œ”(๐ฑ)โ‰ค๐œ”(๐ฒ) in Bruhat order. This follows immediately from the definitionโ€‰โ€‰(2.6).

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

254235.fig.005
, the set ๐ท๐‘ƒ๐‘‡ of top parts of dimension pairs is {2,3,4}, yielding the integer vector ๐ฑ=(1,1,1,0). The associated permutation ๐œ”(๐ฑ) is then ๐‘ 1๐‘ 2๐‘ 3.

Example 2.9. Let ๐œ†,โ„Ž be as in Example 2.5. The filling

254235.fig.006
is also permissible, with dimension pairs (1,2), (1,3), (1,4), (2,3). Hence ๐ฑ=(1,2,1,0) and the associated permutation ๐œ”(๐ฑ) is ๐‘ 1(๐‘ 2๐‘ 1)๐‘ 3.

2.3. Bijection between Fixed Points and Permissible Fillings

For nilpotent Hessenberg varieties, the ๐‘†1-fixed points Hess(๐‘,โ„Ž)๐‘†1 are in bijective correspondence with the set of permissible fillings of the Young diagram ๐œ†=๐œ†๐‘, as we now describe. We will use this correspondence in the formulation of our dimension pair algorithm.

Suppose ๐œ† is a Young diagram with ๐‘› boxes. We begin by defining a bijective correspondence between the set โ„ฑ๐‘–โ„“โ„“(๐œ†) of all fillings (not necessarily permissible) of ๐œ† with permutations in ๐‘†๐‘›. 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 โ„ฑ๐‘–โ„“โ„“(๐œ†)โ†”๐‘†๐‘› is then given by interpreting the resulting word as the one-line notation of a permutation. For example, the filling254235.fig.007(2.7) has associated permutation 641523. It is easily seen that this is a bijective correspondence. Given a filling ๐‘‡ of ๐œ†, we denote its associated permutation by ๐œ™๐œ†(๐‘‡).

Remark 2.10. In the case when ๐‘ is the principal nilpotent ๐‘›ร—๐‘› matrix, the corresponding Young diagram ๐œ†=๐œ†๐‘=(๐‘›) 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 ๐œ™๐œ†โˆ’1(๐‘ค) by just ๐‘ค. For instance, the permissible filling of ๐œ†=(5) in Example 2.9 has associated permutation 43215.
Now let ๐’ซ๐น๐‘–โ„“โ„“(๐œ†,โ„Ž)(2.8) denote the set of (โ„Ž,๐œ†)-permissible fillings of ๐œ†. Recall that elements in Hess(๐‘,โ„Ž)๐‘†1 are viewed as permutations in ๐‘†๐‘› via the identification โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)๐‘‡๐‘›โ‰…๐‘†๐‘›. 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 ๐‘› a positive integer. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function and ๐œ† a Young diagram with ๐‘› boxes. Suppose ๐‘โ„Ž๐‘“ is a nilpotent operator in highest form as chosen in [16] (cf. Remark 2.2) with ๐œ†๐‘โ„Ž๐‘“=๐œ†. Let Hess(๐‘โ„Ž๐‘“,โ„Ž) denote the associated nilpotent Hessenberg variety. Then the map from the ๐‘†1-fixed points Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1 to the set of permissible fillings ๐’ซ๐น๐‘–โ„“โ„“(๐œ†,โ„Ž)๐‘คโˆˆHess(โ„Ž)๐‘†1โŠ†๐‘†๐‘›โŸผ๐œ™๐œ†โˆ’1๎€ท๐‘คโˆ’1๎€ธโˆˆ๐’ซ๐น๐‘–โ„“โ„“(๐œ†,โ„Ž)(2.9) is well defined and is a bijection.

Remark 2.12. In the case when ๐‘ is the principal nilpotent ๐‘›ร—๐‘› matrix, ๐œ† is the Young diagram with only one row. Thus the map (2.9) above simplifies to ๐‘คโ†ฆ๐‘คโˆ’1 where we abuse notation (cf. Remark 2.10) and denote ๐œ™๐œ†โˆ’1(๐‘คโˆ’1) by ๐‘คโˆ’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 ๐‘†๐‘›. 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 ๐‘โ„Ž๐‘“ be a nilpotent ๐‘›ร—๐‘› matrix in highest form chosen as in Remark 2.2 and let ๐œ†โˆถ=๐œ†๐‘โ„Ž๐‘“. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function and Hess(๐‘โ„Ž๐‘“,โ„Ž) the corresponding nilpotent Hessenberg variety.

The definition of the dimension pair algorithm is pure combinatorics. It produces for each Hessenberg fixed point ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1 an element in ๐‘†๐‘›. Following terminology of poset pinball, we denote this function by๐‘Ÿ๐‘œโ„“โ„“โˆถHess๎€ท๐‘โ„Ž๐‘“๎€ธ,โ„Ž๐‘†1โŸถ๐‘†๐‘›.(3.1)

Definition 3.1 (the dimension pair algorithm). We define ๐‘Ÿ๐‘œโ„“โ„“โˆถHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› as follows.(1)Let ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1 and let ๐œ™๐œ†โˆ’1(๐‘คโˆ’1) be its corresponding permissible filling as defined in (2.9).(2)Let ๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1) be the set of dimension pairs in the permissible filling ๐œ™๐œ†โˆ’1(๐‘คโˆ’1).(3)For each โ„“ with 2โ‰คโ„“โ‰ค๐‘›, set ๐‘ฅโ„“|||๎‚†โˆถ=(๐‘Ž,โ„“)โˆฃ(๐‘Ž,โ„“)โˆˆ๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)๎‚‡|||(3.2) as in (2.4) and define ๐ฑโˆถ=(๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›).(4)Define ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆถ=(๐œ”(๐ฑ))โˆ’1 where ๐œ”(๐ฑ) is the permutation associated to the integer vector ๐ฑ defined in (2.6).

Example 3.2. Let ๐œ†, โ„Ž be as in Example 2.5. The permutation ๐‘ค=43215โˆˆ๐‘†๐‘› is in Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1, as can be checked. The associated permissible filling is

254235.fig.008
. In Example 2.9 we saw that the associated permutation is ๐‘ 1(๐‘ 2๐‘ 1)๐‘ 3, so we conclude ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)=๐‘ 3(๐‘ 1๐‘ 2)๐‘ 1.

We next show that the rolldown function ๐‘Ÿ๐‘œโ„“โ„“โˆถHess(โ„Ž)๐‘†1โ†’๐‘†๐‘› 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, Theoremโ€‰โ€‰1.1]). Let ๐‘โ„Ž๐‘“โˆถโ„‚๐‘›โ†’โ„‚๐‘› be a nilpotent matrix in highest form chosen as in Remark 2.2 and let ๐œ†โˆถ=๐œ†๐‘โ„Ž๐‘“. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function and let Hess(๐‘โ„Ž๐‘“,โ„Ž) denote the corresponding nilpotent Hessenberg variety. There is a paving by (complex) affine cells of Hess(๐‘โ„Ž๐‘“,โ„Ž) such that(i)the affine cells are in one-to-one correspondence with Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1, and(ii)the (complex) dimension of the affine cell ๐ถ๐‘ค corresponding to a fixed point ๐‘คโˆˆHess(๐‘,โ„Ž)๐‘†1 isdimโ„‚๎€ท๐ถ๐‘ค๎€ธ=|||๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)|||.(3.3)

In particular, Theorem 3.3 implies that the odd Betti numbers of Hess(๐‘โ„Ž๐‘“,โ„Ž) are 0, and the 2๐‘˜th even Betti number is precisely the number of fixed points ๐‘ค in Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1 such that |๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)|=๐‘˜. Given the regular nilpotent Hessenberg variety Hess(๐‘โ„Ž๐‘“,โ„Ž), denote by ๐‘๐‘˜ its 2๐‘˜th Betti number, that is, ๐‘๐‘˜โˆถ=dimโ„‚๐ป2๐‘˜๎€ทHess๎€ท๐‘โ„Ž๐‘“.,โ„Ž๎€ธ๎€ธ(3.4) We may now formulate the conditions that guarantee that ๐‘Ÿ๐‘œโ„“โ„“โˆถHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› 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)๐‘Ÿ๐‘œโ„“โ„“โˆถHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› is injective,(2)for every ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1, we have ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โ‰ค๐‘ค in Bruhat order, and(3)for every ๐‘˜โ‰ฅ0, ๐‘˜โˆˆโ„ค, we have ๐‘๐‘˜=|||๎‚†๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess๎€ท๐‘โ„Ž๐‘“๎€ธ,โ„Ž๐‘†1with๎‚‡|||,โ„“(๐‘Ÿ๐‘œโ„“โ„“(๐‘ค))=๐‘˜(3.5) where โ„“(๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)) denotes the Bruhat length of roโ„“โ„“(๐‘ค)โˆˆ๐‘†๐‘›.

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(๐‘โ„Ž๐‘“,โ„Ž) is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the function roโ„“โ„“โˆถHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› is injective.

Proof. Since the association ๐ฑ=(๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›)โ†ฆ๐œ”(๐ฑ) given in (2.6) is a bijection, it suffices to show that the map which sends a Hessenberg fixed point ๐‘คโˆˆHess(โ„Ž)๐‘†1 to the list of top parts ๐ฑ 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 ๐‘›โˆ’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 ๐‘คโˆˆHess(โ„Ž)๐‘†1, one has ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โ‰ค๐‘ค in Bruhat order.

Proof. Since Bruhat order is preserved under taking inverses, it suffices to prove that ๐œ”(๐ฑ) is Bruhat-less than ๐‘คโˆ’1. For any permutation ๐‘ขโˆˆ๐‘†๐‘›, set ๐‘ฆโ„“โˆถ={(๐‘Ž,โ„“)โˆฃ(๐‘Ž,โ„“)isaninversionin๐‘ข}(3.6) and let ๐ฒโˆถ=(๐‘ฆ2,๐‘ฆ3,โ€ฆ,๐‘ฆ๐‘›). Then the association (2.6) applied to the vector ๐ฒ recovers the permutation ๐‘ข. By definition of ๐œ™๐œ† and the definition of dimension pairs, the set ๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1) is always a subset of the set of inversions of the permutation ๐‘คโˆ’1. From Fact 1 it follows that the permutation ๐œ”(๐ฑ) is Bruhat-less than ๐‘คโˆ’1 as desired.

Lemma 3.6. Let ๐‘โ„Ž๐‘“โˆถโ„‚๐‘›โ†’โ„‚๐‘› be a nilpotent matrix in highest form chosen as in Remark 2.2 and let ๐œ†โˆถ=๐œ†๐‘โ„Ž๐‘“. Let โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be a Hessenberg function and Hess(๐‘โ„Ž๐‘“,โ„Ž) the associated nilpotent Hessenberg variety. For every ๐‘˜โ‰ฅ0, ๐‘˜โˆˆโ„ค, one has ๐‘๐‘˜=|||๎‚†roโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(โ„Ž)๐‘†1withโ„“(ro๎‚‡|||,โ„“โ„“(๐‘ค))=๐‘˜(3.7) where โ„“(๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)) denotes the Bruhat length of ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆˆ๐‘†๐‘›.

Proof. By construction, ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) has a reduced word decomposition consisting of precisely |๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)| simple transpositions. Hence its Bruhat length is |๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)|. By Theorem 3.3, ๐‘๐‘˜ is precisely the number of fixed points ๐‘ค with |๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)|=๐‘˜ 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(๐‘โ„Ž๐‘“,โ„Ž) is either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Then the association ๐‘คโ†ฆroโ„“โ„“(๐‘ค) given by the dimension pair algorithm is a possible outcome of a successful game of Betti poset pinball played with ambient partially ordered set ๐‘†๐‘› equipped with Bruhat order, rank function ๐œŒ=โ„“โˆถ๐‘†๐‘›โ†’โ„ค given by Bruhat length, initial subset Hess(โ„Ž)๐‘†1โŠ†๐‘†๐‘›, and target Betti numbers ๐‘๐‘˜โˆถ=dimโ„‚๐ป2๐‘˜(Hess(โ„Ž);โ„‚).

Remark 3.8. Lemmas 3.5 and 3.6 hold for general nilpotent ๐‘โ„Ž๐‘“ and Hessenberg functions โ„Ž. 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 ๐‘โ„Ž๐‘“ and โ„Ž.

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

In the context of a GKM-compatible subspace of a GKM space [1, Definitionโ€‰โ€‰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 โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›) with the standard ๐‘‡๐‘›-action and the GKM-compatible subspace is Hess(๐‘โ„Ž๐‘“,โ„Ž) with the ๐‘†1-action specified above. Consider the ๐ปโˆ—๐‘‡๐‘›(pt)-module basis for ๐ปโˆ—๐‘‡๐‘›(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)) given by the equivariant Schubert classes {๐œŽ๐‘ค}๐‘คโˆˆ๐‘†๐‘›. The dimension pair algorithm then specifies the set๎‚†๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess๎€ท๐‘โ„Ž๐‘“๎€ธ,โ„Ž๐‘†1๎‚‡โŠ†๐ปโˆ—๐‘†1๎€ทHess๎€ท๐‘โ„Ž๐‘“,,โ„Ž๎€ธ๎€ธ(3.8) where for any ๐‘ขโˆˆ๐‘†๐‘› the class ๐‘๐‘ขโˆถ=๐œ‹(๐œŽ๐‘ข) is defined to be the image of ๐œŽ๐‘ข under the natural projection map๐œ‹โˆถ๐ปโˆ—๐‘‡๐‘›(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›))โŸถ๐ปโˆ—๐‘†1๎€ทHess๎€ท๐‘โ„Ž๐‘“,โ„Ž๎€ธ๎€ธ(3.9) induced by the inclusion of groups ๐‘†1โ†ช๐‘‡๐‘› and the ๐‘†1-equivariant inclusion of spaces Hess(๐‘โ„Ž๐‘“,โ„Ž)โ†ชโ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). We refer to the images ๐‘๐‘ข as Hessenberg Schubert classes.

Following the methods of [1] we view ๐ปโˆ—๐‘‡๐‘›(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)) and ๐ปโˆ—๐‘†1(Hess(๐‘โ„Ž๐‘“,โ„Ž)) as subrings of๐ปโˆ—๐‘‡๐‘›๎‚€(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›))๐‘‡๐‘›๎‚โ‰…๎ถ๐‘คโˆˆ๐‘†๐‘›๐ปโˆ—๐‘‡๐‘›(pt)respectively๐ปโˆ—๐‘†1๎‚€๎€ทHess๎€ท๐‘โ„Ž๐‘“,โ„Ž๎€ธ๎€ธ๐‘†1๎‚โ‰…๎ถ๐‘คโˆˆHess๎€ท๐‘โ„Ž๐‘“๎€ธ,โ„Ž๐‘†1๐ปโˆ—๐‘†1(pt).(3.10) We denote by ๐œŽ๐‘ค(๐‘คโ€ฒ), ๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)(๐‘คโ€ฒ) the value of the ๐‘คโ€ฒth coordinate in the direct sums above, for ๐‘ค, ๐‘คโ€ฒโˆˆ๐‘†๐‘› or ๐‘ค, ๐‘คโ€ฒโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1, respectively. If๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)(๐‘ค)โ‰ 0,๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)๎€ท๐‘ค๎…ž๎€ธ=0if๐‘คโ‰ฐ๐‘คโ€ฒ(3.11) for all ๐‘ค, ๐‘คโ€ฒโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1, then the set {๐‘roโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1} in ๐ปโˆ—๐‘†1(Hess(๐‘โ„Ž๐‘“,โ„Ž)) is called poset-upper-triangular (with respect to the partial order on Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โŠ†๐‘†๐‘› induced from Bruhat order) [1, Definition 2.3]. Finally, recall that the cohomology degree of an equivariant Schubert class ๐œŽ๐‘ค (and hence also the corresponding Hessenberg Schubert class ๐‘๐‘ค) is 2โ‹…โ„“(๐‘ค).

The following is immediate from [1, Propositionโ€‰โ€‰4.14] and the above discussion.

Proposition 3.9. Let Hess(๐‘โ„Ž๐‘“,โ„Ž) be either a regular nilpotent Hessenberg variety or a nilpotent Springer variety. Let roโ„“โ„“โˆถHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› be the dimension pair algorithm defined above. Suppose (3.11) holds for all ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1. Then the Hessenberg Schubert classes {๐‘roโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1} form a ๐ปโˆ—๐‘†1(pt)-module basis for the ๐‘†1-equivariant cohomology ring ๐ปโˆ—๐‘†1(Hess(๐‘โ„Ž๐‘“,โ„Ž)).

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 ๐‘ค, ๐‘คโ€ฒโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†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 [1, Section 4.3], definedegHess(๐‘โ„Ž๐‘“,โ„Ž)(๐‘ค)โˆถ=dimโ„‚๎€ท๐ถ๐‘ค๎€ธ(3.12) to be the (complex) dimension of the affine cell ๐ถ๐‘ค containing the fixed point ๐‘ค in Tymoczkoโ€™s paving by affines of Hess(๐‘โ„Ž๐‘“,โ„Ž) in Theorem 3.3. Then from the discussion above we knowdegHess(๐‘โ„Ž๐‘“,โ„Ž)|||(๐‘ค)=๐ท๐‘ƒ๐œ™๐œ†โˆ’1(๐‘คโˆ’1)|||=โ„“(๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)),(3.13) and since the cohomology degree of ๐‘roโ„“โ„“(๐‘ค) is 2โ‹…โ„“(๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)), we see that the association ๐‘คโ†ฆ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) from Hess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1โ†’๐‘†๐‘› is also a matching in the sense of [1] with respect to degHess(๐‘โ„Ž๐‘“,โ„Ž) and rank function ๐œŒ on ๐‘†๐‘› given by Bruhat length. Thus the fact that the {๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(๐‘โ„Ž๐‘“,โ„Ž)๐‘†1} form a module basis can also be deduced from [1, Theoremโ€‰โ€‰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 ๐‘†1-equivariant cohomology ring. Here and below the nilpotent operator ๐‘ under consideration is always the principal nilpotent, so we omit the ๐‘ from the notation and write Hess(โ„Ž). Similarly the corresponding Young diagram is always ๐œ†=(๐‘›) so we omit the ๐œ† from notation and write ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž) instead of ๐’ซ๐น๐‘–โ„“โ„“(๐œ†,โ„Ž).

We fix for this discussion the Hessenberg function given byโ„Ž(1)=โ„Ž(2)=3,โ„Ž(๐‘–)=๐‘–+1for3โ‰ค๐‘–โ‰ค๐‘›โˆ’1,โ„Ž(๐‘›)=๐‘›.(4.1) The only difference between this function โ„Ž and the Hessenberg function for the Peterson variety studied in [13] is that the value of โ„Ž(1) is 3 instead of 2. In this sense this โ„Ž is โ€œcloseโ€™โ€™ 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(โ„Ž)๐‘†1 which do not arise in the Peterson case.

The Hessenberg function โ„Ž in (4.1) is trivial if ๐‘›=3 since in that case โ„Ž(1)=โ„Ž(2)=โ„Ž(3)=3 which implies that the corresponding Hessenberv variety Hess(โ„Ž) is equal to the full flag variety โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚3). Hence we assume ๐‘›โ‰ฅ4 throughout. Under this assumption and following the notation introduced in Section 2, the Hessenberg function is of the form โ„Ž=(3,3,4,โ€ฆ). 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 ๐‘›โ‰ฅ4 and let Hess(โ„Ž) be the 334-type Hessenberg variety in โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). Let ๐‘Ÿ๐‘œโ„“โ„“โˆถHess(โ„Ž)๐‘†1โ†’๐‘†๐‘› be the dimension pair algorithm defined in Section 3. Then ๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)(๐‘ค)โ‰ 0,๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)๎€ท๐‘ค๎…ž๎€ธ=0if๐‘คโ‰ฐ๐‘คโ€ฒ(4.2) for all ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1. In particular the Hessenberg Schubert classes {๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(โ„Ž)๐‘†1} form a ๐ปโˆ—๐‘†1(pt)-module basis for the ๐‘†1-equivariant cohomology ring ๐ปโˆ—๐‘†1(Hess(โ„Ž)).

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 ๐‘›, โ„Ž, Hess(โ„Ž) and ๐‘Ÿ๐‘œโ„“โ„“ be as above. Then, ๐‘roโ„“โ„“(๐‘ค)(๐‘ค)โ‰ 0(4.3) for all ๐‘คโˆˆHess(โ„Ž)๐‘†1.

Proposition 4.3. Let ๐‘›, โ„Ž, Hess(โ„Ž) and ๐‘Ÿ๐‘œโ„“โ„“ be as above. Then, ๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)๎€ท๐‘ค๎…ž๎€ธ=0if๐‘คโ‰ฐ๐‘คโ€ฒ(4.4) for all ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1.

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 ๐‘›, โ„Ž, Hess(โ„Ž) and ๐‘Ÿ๐‘œโ„“โ„“ be as above. If for all ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1 one has ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โ‰ค๐‘คโ€ฒโŸบ๐‘คโ‰ค๐‘คโ€ฒ(4.5) in Bruhat order, then the Hessenberg Schubert classes {๐‘๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โˆฃ๐‘คโˆˆHess(โ„Ž)๐‘†1} satisfy (4.4).

Proof. Recall that the equivariant Schubert classes are poset-upper-triangular with respect to Bruhat order on ๐‘†๐‘›. In particular, for all ๐‘ค, ๐‘คโ€ฒโˆˆ๐‘†๐‘› we have ๐œŽ๐‘ค(๐‘คโ€ฒ)=0 if ฬธ๐‘คโ€ฒโ‰ฅ๐‘ค. Since the Hessenberg Schubert classes are images of the Schubert classes and the diagram ๐ปโˆ—๐‘‡๐‘›(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›))๐ปโˆ—๐‘‡๐‘›๎‚€(โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›))๐‘‡๐‘›๎‚โ‰…๎ถ๐‘คโˆˆ๐‘Š๐ปโˆ—๐‘‡๐‘›(pt)๐ปโˆ—๐‘†1(Hess(โ„Ž))๐ปโˆ—๐‘†1๎‚€(Hess(โ„Ž))๐‘†1๎‚โ‰…๎ถ๐‘คโˆˆHess(โ„Ž)๐‘†1๐ปโˆ—๐‘†1(pt)(4.6) commutes, it follows that if for all ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1, we have ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โ‰ค๐‘คโ€ฒโŸบ๐‘คโ‰ค๐‘คโ€ฒ(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(โ„Ž)๐‘†1 a subset of {1,2,โ€ฆ,๐‘›โˆ’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 ๐‘›โˆˆโ„• and let โ„Ž,โ„Žโ€ฒโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›} be two Hessenberg functions. If โ„Ž(๐‘–)โ‰ฅโ„Žโ€ฒ(๐‘–) for all ๐‘–, 1โ‰ค๐‘–โ‰ค๐‘›, then Hess๎€ทโ„Ž๎…ž๎€ธโŠ†Hess(โ„Ž).(4.8) The inclusion Hess(โ„Žโ€ฒ)โ†ชHess(โ„Ž) is ๐‘†1-equivariant and in particular Hess(โ„Žโ€ฒ)๐‘†1โŠ†Hess(โ„Ž)๐‘†1 and ๐’ซ๐น๐‘–โ„“โ„“(โ„Žโ€ฒ)โŠ†๐’ซ๐น๐‘–โ„“โ„“(โ„Ž).

Proof. Let ๐‘‰โ€ข=(๐‘‰๐‘–) denote an element in โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). By definition the regular nilpotent Hessenberg variety Hess(โ„Žโ€ฒ) associated to โ„Žโ€ฒ is Hess๎€ทโ„Ž๎…ž๎€ธ๎€ฝ๐‘‰โˆถ=โ€ขโˆˆโ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›)โˆฃ๐‘๐‘‰๐‘–โŠ†๐‘‰โ„Žโ€ฒ(๐‘–)๎€พ,,โˆ€1โ‰ค๐‘–โ‰ค๐‘›(4.9) where ๐‘ is the principal nilpotent operator. Since ๐‘‰๐‘–โŠ†๐‘‰๐‘–+1 for all 1โ‰ค๐‘–โ‰ค๐‘›โˆ’1 by definition of flags and ๐‘‰๐‘›=โ„‚๐‘› for all flags, if โ„Žโ€ฒ(๐‘–)โ‰คโ„Ž(๐‘–) for all ๐‘–, then ๐‘๐‘‰๐‘–โŠ†๐‘‰โ„Ž๎…ž(๐‘–) automatically implies ๐‘๐‘‰๐‘–โŠ†๐‘‰โ„Ž(๐‘–). We conclude Hess(โ„Žโ€ฒ)โŠ†Hess(โ„Ž). The ๐‘†1-equivariance of the inclusion Hess(โ„Žโ€ฒ)โ†ชHess(โ„Ž) follows from the definition of the ๐‘†1-action of (2.3).

Applying Lemma 4.5 to the Hessenberg functionโ„Ž๎…ž(๐‘–)=๐‘–+1for1โ‰ค๐‘–โ‰ค๐‘›โˆ’1,โ„Ž๎…ž(๐‘›)=๐‘›(4.10) corresponding to the Peterson variety Hess(โ„Žโ€ฒ) and โ„Ž the 334-type Hessenberg function (4.1), we conclude that all fixed points in Hess(โ„Žโ€ฒ)๐‘†1 also arise as fixed points in Hess(โ„Ž)๐‘†1. We refer to the elements of Hess(โ„Žโ€ฒ)๐‘†1 (viewed as elements of Hess(โ„Ž)๐‘†1) as Peterson-type fixed points. It therefore remains to describe Hess(โ„Ž)๐‘†1โงตHess(โ„Žโ€ฒ)๐‘†1. It turns out to be convenient to do this by first describing ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž)โงต๐’ซ๐น๐‘–โ„“โ„“(โ„Žโ€ฒ).

We first introduce some terminology. Given a permutation ๐‘ค=(๐‘ค(1)๐‘ค(2)โ‹ฏ๐‘ค(๐‘›)) in one-line notation and some ๐‘–, โ„“, we say that the entries {๐‘ค(๐‘–),๐‘ค(๐‘–+1),โ€ฆ,๐‘ค(๐‘–+โ„“)} form a decreasing staircase, or simply a staircase, if ๐‘ค(๐‘—+1)=๐‘ค(๐‘—)โˆ’1 for all ๐‘–โ‰ค๐‘—<๐‘–+โ„“. For example, for ๐‘ค=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, ๐‘ค=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, ๐‘ค=321654987 is an increasing sequence of (three) staircases 321, 654, and 987.

It is shown in [13] that the ๐‘†1-fixed points of the Peterson variety Hess(โ„Žโ€ฒ) consist precisely of those permutations ๐‘คโˆˆ๐‘†๐‘› such that the one-line notation of ๐‘ค is an increasing sequence of staircases. Since such ๐‘ค are equal to their own inverses, the permissible fillings ๐’ซ๐น๐‘–โ„“โ„“(โ„Žโ€ฒ) corresponding to Hess(โ„Žโ€ฒ) are precisely those which are increasing sequences of staircases (cf. Remark 2.12). We now describe the permissible fillings ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž) which are not Peterson-type fillings. We use the language of โ„Ž-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 ๐‘†๐‘› via one-line notation.

Lemma 4.6. Let ๐‘›โ‰ฅ4 and let Hess(โ„Ž) be the 334-type Hessenberg variety in โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). Let ๐‘คโˆˆ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž) be a permissible filling for Hess(โ„Ž) which is not of Peterson type, that is, ๐‘คโˆˆ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž)โงต๐’ซ๐น๐‘–โ„“โ„“(โ„Žโ€ฒ). Then precisely one of the following hold.(i)The one-line notation of ๐‘ค is of the form ๐‘ค๎…ž๐‘ค๎…ž312๐‘ค๎…ž๎…ž,(4.11) where ๐‘คโ€ฒ is a (possibly empty) staircase such that ๐‘ค๎…ž3 is also a staircase, and ๐‘คโ€ฒโ€ฒ is an increasing sequence of staircases. We refer to these as 312-type permissible fillings. (ii)The one-line notation of ๐‘ค is of the form 2๐‘ค๎…ž31๐‘ค๎…ž๎…ž,(4.12) where ๐‘ค๎…ž is a (possibly empty) staircase such that ๐‘ค๎…ž3 is also a staircase, and ๐‘ค๎…ž๎…ž 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 ๐’ซ๐น๐‘–โ„“โ„“(โ„Ž)โงต๐’ซ๐น๐‘–โ„“โ„“(โ„Žโ€ฒ).

Proof of Lemma 4.6. For any Hessenberg function โ„Žโˆถ{1,2,โ€ฆ,๐‘›}โ†’{1,2,โ€ฆ,๐‘›}, Mbirika shows in [14, Section 3.2] that the Level ๐‘› fillings in an โ„Ž-tableau tree are precisely the permissible fillings with respect to โ„Ž. For the Peterson Hessenberg function in (4.10) Mbirikaโ€™s corresponding โ„Ž-tableau tree has the property that for every ๐‘˜ with 1โ‰ค๐‘˜โ‰ค๐‘›โˆ’1 and every vertex at Level ๐‘˜, there are precisely 2 edges going down from that vertex to a Level ๐‘˜+1 vertex (this is because the corresponding degree tuple ๐›ฝ [14, Definitionโ€‰โ€‰3.1.1] has ๐›ฝ๐‘–=2 for all 1โ‰ค๐‘–โ‰ค๐‘›โˆ’1). In the case of the 334-type Hessenberg function, by definition the โ„Ž-tableau tree also has precisely 2 edges going down from every vertex at Level ๐‘˜ for all ๐‘˜โ‰ 2, 1โ‰ค๐‘˜โ‰ค๐‘›โˆ’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, Definitionโ€‰3.1.9]) it can be seen that for the case of the Peterson Hessenberg function, the corresponding โ„Ž-tableau tree at Level 2 has vertices โ€ข21โ€ข and 1โ€ข2โ€ข, whereas for the 334-type Hessenberg function, the Level 2 vertices have the form โ€ข2โ€ข1โ€ข and โ€ข1โ€ข2โ€ข. Here the bullets indicate the locations of the โ„Ž-permissible positions available for the placement of the next index 3, in the sense of [14, Section 3] (cf. in particular [14, Lemmaโ€‰โ€‰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 321โ€ข, 213โ€ข (coming from โ€ข2โ€ข1โ€ข) and the two vertices 1โ€ข32โ€ข, 12โ€ข3โ€ข (coming from โ€ข1โ€ข2โ€ข) are identical to the corresponding branches in the โ„Ž-tableau tree for the Peterson Hessenberg function. Hence all permissible fillings at the final Level ๐‘› of these branches are of Peterson type. In contrast, the branches emanating from 2โ€ข31โ€ข and โ€ข312โ€ข do not appear in the Peterson โ„Ž-tableau tree, and none of the fillings appearing at Level ๐‘› 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 โ„Ž-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 ๐‘›=8. Then ๐‘ค=54312876 is an example of a 312-type permissible filling where ๐‘คโ€ฒ=54 and ๐‘ค๎…ž๎…ž=876. An example of a 231-type permissible filling is ๐‘ค=25431876 where ๐‘คโ€ฒ=54 and ๐‘ค๎…ž๎…ž=876. Neither of these are permissible with respect to the Peterson Hessenberg function โ„Žโ€ฒ 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 ๐‘ค=54321876; this relationship is closely analyzed and used below.

We now give explicit descriptions of the corresponding non-Peterson-type elements in Hess(โ„Ž)๐‘†1, obtained by taking inverses of the permissible fillings described in Lemma 4.6.

Definition 4.8. Let ๐‘คโˆˆHess(โ„Ž)๐‘†1. We say ๐‘ค is a 312-type (resp., 231-type) fixed point if its inverse ๐‘คโˆ’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 ๐‘ค be a 312-type (resp., 231-type) permissible filling. Let ๐‘Ž2 be the integer such that ๐‘Ž2+1 is the first entry (resp., second entry) in the one-line notation of ๐‘ค. Let ๐‘คโˆ’1 be the corresponding 312-type (resp., 231 type) fixed point. Then,(i)the one-line notation of ๐‘คโˆ’1 is the same as that of ๐‘ค for all โ„“th entries with โ„“>๐‘Ž2+1,(ii)if ๐‘ค is 312-type, then the first ๐‘Ž2+1 entries of the one-line notation of ๐‘คโˆ’1 are ๐‘Ž2๐‘Ž2+1๐‘Ž2โˆ’1๐‘Ž2โˆ’2โ‹ฏ21,(4.13)(iii)if ๐‘ค is 231-type, then the first ๐‘Ž2+1 entries of the one-line notation of ๐‘คโˆ’1 are ๐‘Ž2+11๐‘Ž2๐‘Ž2โˆ’1โ‹ฏ32.(4.14)

In the case of the Peterson variety, there is a convenient bijective correspondence between the set of ๐‘†1-fixed points of the Peterson variety and subsets ๐’œ of {1,2,โ€ฆ,๐‘›โˆ’1} given as follows [13, Section 2.3]. Let ๐‘ค be a Peterson-type fixed point. Then the corresponding subset is๐’œโˆถ={๐‘–โˆถ1โ‰ค๐‘–โ‰ค๐‘›โˆ’1,๐‘ค(๐‘–)=๐‘ค(๐‘–+1)+1}โŠ†{1,2,โ€ฆ,๐‘›โˆ’1}.(4.15) In the case of the 334-type Hessenber variety, it is also useful to assign a subset of {1,2,โ€ฆ,๐‘›โˆ’1} to each fixed point as follows.

Definition 4.10. Let ๐‘คโˆˆHess(โ„Ž)๐‘†1. The associated subset of {1,2,โ€ฆ,๐‘›}corresponding to ๐‘ค, notated ๐’œ(๐‘ค), is defined as follows.(i)Suppose ๐‘ค is of Peterson type. Then ๐’œ(๐‘ค) is defined to be the set ๐’œ in (4.15).(ii)Suppose ๐‘ค is 312-type. Consider the permutation ๐‘คโ€ฒโˆถ=๐‘ค๐‘ 1 (i.e., swap the ๐‘Ž2 and the ๐‘Ž2+1 in the one-line notation (4.13)). This is a fixed point of Peterson type. Define ๐’œ(๐‘ค)โˆถ=๐’œ(๐‘คโ€ฒ).(iii)Suppose ๐‘ค is 231-type. Consider the permutation ๐‘คโ€ฒ=๐‘ค๐‘ 2๐‘ 3โ‹ฏ๐‘ ๐‘Ž2(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. โ€‰Define ๐’œ(๐‘ค)โˆถ=๐’œ(๐‘คโ€ฒ).

Example 4.11. Suppose ๐‘›=8.(i)Suppose ๐‘ค is the Peterson-type fixed point ๐‘ค=54321876. Then ๐’œ(๐‘ค)={1,2,3,4}โˆช{6,7}. This agrees with the association ๐‘คโ†ฆ๐’œ(๐‘ค) used in [13].(ii)Suppose ๐‘ค is the 312-type fixed point ๐‘ค=34217658 (corresponding to the 312-type permissible filling 43127658). Then ๐‘คโ€ฒ=๐‘ค๐‘ 1=43217658 and ๐’œ(๐‘ค)โˆถ=๐’œ(๐‘คโ€ฒ)={1,2,3}โˆช{5,6}.(iii)Suppose ๐‘ค is the 231-type fixed point ๐‘ค=51432768 (corresponding to the 231-type permissible filling 25431768). Then ๐‘คโ€ฒ=54321768 and ๐’œ(๐‘ค)โˆถ=๐’œ(๐‘คโ€ฒ)={1,2,3,4}โˆช{6}.

Remark 4.12. The three fixed points ๐‘ค=54321876, ๐‘ค=45321876, and ๐‘ค=51432876, which are, respectively, of Peterson type, 312 type, and 231-type, all have the same associated subset ๐’œ(๐‘ค)={1,2,3,4}โˆช{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 ๐‘ค be a 334-type Hessenberg fixed point. Suppose further that ๐‘ค is not of Peterson type. Then {1,2}โŠ†๐’œ(๐‘ค).

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 ๐‘Ž2๐‘Ž2+1โ‹ฏ21 (resp, ๐‘Ž2+11๐‘Ž2โ‹ฏ32) in the one-line notation is such that ๐‘Ž2โ‰ฅ2. From Definition 4.10 it follows that the first decreasing staircase of the associated Peterson-type fixed point ๐‘ค๐‘ 1 (resp., ๐‘ค๐‘ 2๐‘ 3โ‹ฏ๐‘ ๐‘Ž2) is of length at least 3. In particular, the first staircase starts with an integer ๐‘˜ which is โ‰ฅ 3. The result follows.

As noted in Remark 4.12, the association ๐‘คโ†ฆ๐’œ(๐‘ค) 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(โ„Ž)๐‘†1 which we use in our arguments below. We also compute, and fix choices of reduced words for, the rolldowns ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) of the fixed points.

The association ๐‘คโ†ฆ๐’œ(๐‘ค) of the previous section allows us to describe these reduced word decompositions in relation to that of the Peterson-type fixed points. Let ๐‘Ž be a positive integer and ๐‘˜ a nonnegative integer. Recall that a reduced word decomposition of the maximal element (the full inversion) in the subgroup ๐‘†{๐‘Ž,๐‘Ž+1,โ€ฆ,๐‘Ž+๐‘˜+1}โŠ†๐‘†๐‘› is given by๐‘ ๐‘Ž๎€ท๐‘ ๐‘Ž+1๐‘ ๐‘Ž๐‘ ๎€ธ๎€ท๐‘Ž+2๐‘ ๐‘Ž+1๐‘ ๐‘Ž๎€ธโ‹ฏ๎€ท๐‘ ๐‘Ž+๐‘˜๐‘ ๐‘Ž+๐‘˜โˆ’1โ‹ฏ๐‘ ๐‘Ž+1๐‘ ๐‘Ž๎€ธ.(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 {๐‘Ž,๐‘Ž+1,โ€ฆ,๐‘Ž+๐‘˜} for ๐‘Ž positive and ๐‘˜ a nonnegative integer by [๐‘Ž,๐‘Ž+๐‘˜]. We say that [๐‘Ž,๐‘Ž+๐‘˜] is a maximal consecutive substring of ๐’œ if [๐‘Ž,๐‘Ž+๐‘˜]โŠ†๐’œ and neither ๐‘Žโˆ’1 nor ๐‘Ž+๐‘˜+1 are in ๐’œ. It is straightforward that any subset ๐’œ of {1,2,โ€ฆ,๐‘›โˆ’1} uniquely decomposes into a disjoint union of maximal consecutive substrings๎€บ๐‘Ž๐’œ=1,๐‘Ž2๎€ปโˆช๎€บ๐‘Ž3,๐‘Ž4๎€ป๎€บ๐‘Žโˆชโ‹ฏโˆช๐‘šโˆ’1,๐‘Ž๐‘š๎€ป.(4.18) For instance, for ๐’œ={1,2,3,5,6,9,10,11}, the decomposition is ๐’œ=[1,3]โˆช[5,6]โˆช[9,11]. For any [๐‘Ž,๐‘], denote by ๐‘ค[๐‘Ž,๐‘] the full inversion in the subgroup ๐‘†[๐‘Ž,๐‘+1]. Then it follows from Definition 4.10 (see also [13, Section 2.3]) that the Peterson-type fixed point associated to ๐’œ, which we denote by ๐‘ค๐’œ, is the product๐‘ค๐’œโˆถ=๐‘ค[๐‘Ž1,๐‘Ž2]๐‘ค[๐‘Ž3,๐‘Ž4]๐‘ค[๐‘Ž5,๐‘Ž6]โ‹ฏ๐‘ค[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š].(4.19) We fix a choice of reduced word decomposition of ๐‘ค๐’œ given by taking the product of the standard reduced words (4.17) for each of the full inversions ๐‘ค[๐‘Ž๐‘—,๐‘Ž๐‘—+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 ๐‘ค๐’œ.

Example 4.14. Let ๐‘›=7 and let ๐‘ค=4321765 be a Peterson-type fixed point. Then the two decreasing staircases are 4321 and 765, the associated subset ๐’œ(๐‘ค) is {1,2,3}โˆช{5,6} with maximal consecutive strings [1,3]โˆถ={1,2,3} and [5,6]โˆถ={5,6}. The standard reduced word decomposition of ๐‘ค is ๐‘ค{1,2,3}โˆช{5,6}=๐‘ค[1,3]๐‘ค[5,6]=๐‘ 1๎€ท๐‘ 2๐‘ 1๐‘ ๎€ธ๎€ท3๐‘ 2๐‘ 1๎€ธ๐‘ 5๎€ท๐‘ 6๐‘ 5๎€ธ.(4.20)

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

Lemma 4.15. Let ๐‘คโˆˆHess(โ„Ž)๐‘†1 be a fixed point which is not of Peterson type and let ๐’œ(๐‘ค)=[๐‘Ž1,๐‘Ž2]โˆช[๐‘Ž3,๐‘Ž4]โˆชโ‹ฏโˆช[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š] be the associated subset with its decomposition into maximal consecutive substrings.(i)If ๐‘ค is 312-type, then a reduced word decomposition for ๐‘ค is given by ๐‘ 1๎€ท๐‘ 2๐‘ 1๎€ธโ‹ฏ๎€ท๐‘ ๐‘Ž2๐‘ ๐‘Ž2โˆ’1โ‹ฏ๐‘ 3๐‘ 2๎€ธ๐‘ค[๐‘Ž3,๐‘Ž4]โ‹ฏ๐‘ค[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š].(4.21)(ii)If ๐‘ค is 231-type, then a reduced word decomposition for ๐‘ค is given by ๐‘ 2๎€ท๐‘ 3๐‘ 2๎€ธโ‹ฏ๎€ท๐‘ ๐‘Ž2โˆ’1๐‘ ๐‘Ž2โˆ’2โ‹ฏ๐‘ 3๐‘ 2๐‘ ๎€ธ๎€ท๐‘Ž2๐‘ ๐‘Ž2โˆ’1โ‹ฏ๐‘ 2๐‘ 1๎€ธ๐‘ค[๐‘Ž3,๐‘Ž4]โ‹ฏ๐‘ค[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š],(4.22) where the ๐‘ค[๐‘Žโ„“,๐‘Žโ„“+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 ๐‘ค has precisely 1 fewer inversion than ๐‘ค๐’œ(๐‘ค). An explicit computation shows that the given word (4.21) is equal to ๐‘ค, so it is a word decomposition of ๐‘ค with exactly as many simple transpositions as the Bruhat length of ๐‘ค. In particular it must be reduced. A similar argument proves the second assertion.

Example 4.16. Suppose ๐‘›=7. Suppose ๐‘ค=3421765 is a 312-type fixed point. Then the reduced word decomposition of ๐‘ค given in Lemma 4.15 is ๐‘ค=๐‘ 1๎€ท๐‘ 2๐‘ 1๐‘ ๎€ธ๎€ท3๐‘ 2๎€ธ๐‘ 5๎€ท๐‘ 6๐‘ 5๎€ธ.(4.23) Similarly suppose ๐‘ค=4132765 is a 231-type fixed point. Then the reduced word decomposition of ๐‘ค given in Lemma 4.15 is ๐‘ค=๐‘ 2๎€ท๐‘ 3๐‘ 2๐‘ 1๎€ธ๐‘ 5๎€ท๐‘ 6๐‘ 5๎€ธ.(4.24)

Henceforth, we always use the reduced words given above.

Next we explicitly describe the rolldowns ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) associated to each ๐‘ค in Hess(โ„Ž)๐‘†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 ๐‘คcontains the string 321 (or simply contains 321) if, in the one-line notation of ๐‘ค, the string 321 appears (equivalently, if {1,2}โŠ†๐’œ(๐‘ค)). We say ๐‘คdoes 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 ๐‘ค to contain the string 321 exactly.

Given a subset ๐’œ={๐‘—1<๐‘—2<โ‹ฏ<๐‘—๐‘˜}โŠ†{1,2,โ€ฆ,๐‘›โˆ’1} and corresponding Peterson-type fixed point ๐‘ค๐’œ, we call the permutation๐‘ ๐‘—๐‘˜๐‘ ๐‘—๐‘˜โˆ’1โ‹ฏ๐‘ ๐‘—2๐‘ ๐‘—1โˆˆ๐‘†๐‘›(4.25) the Peterson case rolldown of ๐‘ค๐’œ. 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, Definitionโ€‰โ€‰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, Propositionโ€‰โ€‰5.16]).

Lemma 4.19. Let ๐‘›โ‰ฅ4 and let Hess(โ„Ž) be the 334-type Hessenberg variety in โ„ฑโ„“๐‘Ž๐‘”๐‘ (โ„‚๐‘›). Let ๐‘ค be a Peterson-type fixed point and let ๐’œ(๐‘ค)={๐‘—1<๐‘—2<โ‹ฏ<๐‘—๐‘˜} be its associated subset.(i)Suppose ๐‘ค does not contain 321. Then ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) is the Peterson case rolldown of ๐‘ค๐’œ(๐‘ค).(ii)Suppose ๐‘ค does contain 321, that is, ๐’œ(๐‘ค)={๐‘—1<๐‘—2<โ‹ฏ<๐‘—๐‘˜} for ๐‘˜โ‰ฅ2 and ๐‘—1=1 and ๐‘—2=2. Then ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) is๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)=๐‘ ๐‘—๐‘˜๐‘ ๐‘—๐‘˜โˆ’1โ‹ฏ๐‘ ๐‘—3๐‘ 1๐‘ 2๐‘ 1.(4.26) In particular, if a Peterson-type fixed point ๐‘ค contains 321, then its rolldown ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) is Bruhat-greater and has Bruhat length 1 greater than the Peterson case rolldown of ๐‘ค.

Proof. If ๐‘ค contains a 321, then by Definition 2.6, the pairs (1,3),(2,3), and (1,2) are all dimension pairs in ๐‘ค. 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 ๐œ”(๐ฑ) begins with the word ๐‘ 1(๐‘ 2๐‘ 1). With respect to all other indices ๐‘—โˆˆ๐’œ(๐‘ค), the 334-type Hessenberg function is identical to the Peterson Hessenberg function and hence, for each such ๐‘—, the index ๐‘—+1 appears precisely once as a top part of a dimension pair of ๐‘ค and thus contributes precisely one ๐‘ ๐‘— to ๐œ”(๐ฑ). Taking the inverse yields (4.26) as desired.
If ๐‘ค does not contain 321, then 3 appears at most once as the top part of a dimension pair in ๐‘ค, and again for all other indices the computations are identical to the Peterson case as above. Hence ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) 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 ๐‘คโˆˆHess(โ„Ž)๐‘†1 and suppose that ๐‘ค is not of Peterson type. Let ๐’œ(๐‘ค)={๐‘—1=1<๐‘—2=2<๐‘—3<โ‹ฏ<๐‘—๐‘˜} for some ๐‘˜โ‰ฅ2.(1)If ๐‘ค is of 312-type, then the dimension pair algorithm associates to ๐‘ค the permutation ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)=๐‘ ๐‘—๐‘˜๐‘ ๐‘—๐‘˜โˆ’1โ‹ฏ๐‘ ๐‘—4๐‘ ๐‘—3๐‘ 1๐‘ 2.(4.27)(2)If ๐‘ค is 231-type, then the dimension pair algorithm associates to ๐‘ค the permutation๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)=๐‘ ๐‘—๐‘˜๐‘ ๐‘—๐‘˜โˆ’1โ‹ฏ๐‘ ๐‘—4๐‘ ๐‘—3๐‘ 2๐‘ 1.(4.28)

Proof. Suppose ๐‘ค is a 312-type fixed point so ๐œ™๐œ†โˆ’1(๐‘คโˆ’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 ๐’œ(๐‘ค) imply that the other dimension pairs are precisely the pairs (๐‘—,๐‘—+1) for ๐‘—โˆˆ๐’œ(๐‘ค) (for ๐‘—โ‰ 1,2), from which it follows that ๐œ”(๐ฑ)=๐‘ 2๐‘ 1๐‘ ๐‘—3๐‘ ๐‘—4โ‹ฏ๐‘ ๐‘—๐‘˜โˆ’1๐‘ ๐‘—๐‘˜. Taking inverses yields (4.27). The proof of the second assertion is similar.

Example 4.21. (i) Suppose ๐‘ค=54321876. This is of Peterson type. Then roโ„“โ„“(๐‘ค)=(๐œ”(๐ฑ))โˆ’1=๐‘ 7๐‘ 6๐‘ 4๐‘ 3๐‘ 1๐‘ 2๐‘ 1.(ii)Suppose ๐‘ค=45321876. This is 312-type. Then roโ„“โ„“(๐‘ค)=(๐œ”(๐ฑ))โˆ’1=๐‘ 7๐‘ 6๐‘ 4๐‘ 3๐‘ 1๐‘ 2.(iii)Suppose ๐‘ค=51432876. This is 231-type. Then roโ„“โ„“(๐‘ค)=(๐œ”(๐ฑ))โˆ’1=๐‘ 7๐‘ 6๐‘ 4๐‘ 3๐‘ 2๐‘ 1.

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 ๐‘ค be a 334-type Hessenberg fixed point and let ๐’œ(๐‘ค)=[๐‘Ž1,๐‘Ž2]โˆชโ‹ฏโˆช[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š] be its associated subset with its decomposition into maximal consecutive substrings. Suppose ๐‘ค is of Peterson type that contains 321, 312-type, or of 231 type. Then ๐‘Ž1=1, ๐‘Ž2โ‰ฅ2 and the first ๐‘Ž2+1 entires of the one-line notation of ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) are ๐‘Ž2+12134โ‹ฏ๐‘Ž2(4.29) for ๐‘ค of Peterson type that contains 321, 2๐‘Ž2+1134โ‹ฏ๐‘Ž2(4.30) for ๐‘ค312-type, and ๐‘Ž2+1123โ‹ฏ๐‘Ž2(4.31) for ๐‘ค231-type.

4.3. Bruhat Order Relations

In this section we analyze the properties of the association ๐‘คโ†ฆ๐’œ(๐‘ค) with respect to comparisons in Bruhat order.

The first two lemmas are straightforward and proofs are left to the reader.

Lemma 4.23. Let ๐’œโŠ†{1,2,โ€ฆ,๐‘›โˆ’1} and let ๐‘ค๐’œ be the Peterson-type filling associated to ๐’œ. Then ๐‘ค๐’œ is maximal in the subgroup ๐‘†๐’œ of ๐‘†๐‘› generated by the simple transpositions {๐‘ ๐‘–}๐‘–โˆˆ๐’œ. In particular, ๐‘ค๐’œ is Bruhat-bigger than any permutation ๐‘คโˆˆ๐‘†๐’œ.

Lemma 4.24. Let ๐‘คโˆˆHess(โ„Ž)๐‘†1. Suppose ๐‘ค is not of Peterson type. Then ๐‘ค is Bruhat-less than the Peterson type fixed point ๐‘ค๐’œ(๐‘ค) corresponding to ๐’œ(๐‘ค).

We also observe that a Bruhat relation ๐‘ค<๐‘คโ€ฒ implies a containment relation of the associated subsets.

Lemma 4.25. Let ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1 and let ๐’œ(๐‘ค), ๐’œ(๐‘คโ€ฒ) be the respective associated subsets. Let ๐‘ ๐‘– be a simple transposition. Then,(1)๐‘ ๐‘–<๐‘ค if and only if ๐‘–โˆˆ๐’œ(๐‘ค),(2)๐‘ ๐‘–<๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) if and only if ๐‘–โˆˆ๐’œ(๐‘ค),(3)if ๐‘คโ‰ค๐‘คโ€ฒ or ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)โ‰ค๐‘ค๎…ž, then ๐’œ(๐‘ค)โŠ†๐’œ(๐‘คโ€ฒ).

Proof. Bruhat order is independent of choice of reduced word decomposition for ๐‘ค. Therefore, a simple transposition ๐‘ ๐‘– is less than ๐‘ค in Bruhat order if and only if ๐‘ ๐‘– appears in a (and hence any) reduced word decomposition of ๐‘ค. In particular, to prove the first claim it suffices to observe that by the definitions of ๐’œ(๐‘ค), the index ๐‘– appears in ๐’œ(๐‘ค) precisely when ๐‘ ๐‘– appears in the choice of reduced word for ๐‘ค given above. A similar argument using the explicit reduced words given for ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค) in Lemmas 4.19 and 4.20 proves the second claim. The last claim follows from the first two.

We have just seen that ๐‘คโ‰ค๐‘คโ€ฒ implies ๐’œ(๐‘ค)โŠ†๐’œ(๐‘คโ€ฒ). In the case of the Peterson variety Hess(โ„Žโ€ฒ) these Bruhat relations are precisely encoded by the partial ordering given by containment of the ๐’œ(๐‘ค); specifically, by Lemma 4.23, ๐‘ค๐’œโ‰ค๐‘คโ„ฌ if and only if ๐’œโŠ†โ„ฌ. In our 334-type Hessenberg case this is no longer true although the sets ๐’œ(๐‘ค) 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 ๐‘†๐‘› (see, e.g., [17]) which will be useful in the discussion below. For ๐‘คโˆˆ๐‘†๐‘›, denote by ๐ท๐‘…(๐‘ค) the descent set of ๐‘ค, namely,๐ท๐‘…(๐‘ค)โˆถ={๐‘–โˆฃ๐‘ค(๐‘–)>๐‘ค(๐‘–+1),1โ‰ค๐‘–โ‰ค๐‘›โˆ’1}.(4.32) For example, for ๐‘ค=368475912 the descent set is ๐ท๐‘…(๐‘ค)={3,5,7}.

Theorem 4.26 (the tableau criterion [17, Theoremโ€‰โ€‰2.6.3]). For ๐‘ค, ๐‘ฃโˆˆ๐‘†๐‘›, let ๐‘ค๐‘–,๐‘˜ be the ๐‘–th element in the increasing rearrangement of ๐‘ค(1),๐‘ค(2),โ€ฆ,๐‘ค(๐‘˜), and similarly for ๐‘ฃ๐‘–,๐‘˜. Then ๐‘คโ‰ค๐‘ฃ in Bruhat order if and only if ๐‘ค๐‘–,๐‘˜โ‰ค๐‘ฃ๐‘–,๐‘˜,โˆ€๐‘˜โˆˆ๐ท๐‘…(๐‘ค),1โ‰ค๐‘–โ‰ค๐‘˜.(4.33)

For example, suppose ๐‘ค=368475912 and ๐‘ฃ=694287531. Since ๐ท๐‘…(๐‘ค)={3,5,7}, we examine the three increasing rearrangements of initial segments of ๐‘ค and ๐‘ฃ of lengths 3, 5, and 7, respectively, which we may organize into Young tableaux: 254235.fig.0010(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 ๐‘คฬธ<๐‘ฃ.

Now we observe that some Bruhat relations never arise.

Lemma 4.27. Let ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1. Let ๐’œ(๐‘ค)=[๐‘Ž1,๐‘Ž2]โˆชโ‹ฏโˆช[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š] be the associated subset of ๐‘ค with its decomposition into maximal consecutive substrings. Suppose one of the following conditions hold:(1)๐‘คโ€ฒ is of Peterson type that does not contain 321 while ๐‘ค is not,(2)๐‘คโ€ฒ is 231-type while ๐‘ค is either of Peterson type that contains 321 or is 312-type. Then ๐‘คฬธ<๐‘คโ€ฒ and ฬธ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)<๐‘คโ€ฒ.

Proof. If ๐‘คโ€ฒ is of Peterson type that does not contain 321, then ฬธ{1,2}โŠ†๐’œ(๐‘คโ€ฒ) 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 ๐’œ(๐‘ค). The claim (1) now follows from Lemma 4.25.
Next suppose ๐‘คโ€ฒ is 231-type and ๐‘ค is of Peterson type that contains 321. Then the first two entries of the one-line notation of ๐‘ค must be both strictly greater than 1, and 2โˆˆ๐ท๐‘…(๐‘ค). Similarly if ๐‘ค is a 312-type fixed point, then ๐‘Ž2โ‰ฅ2. From (4.13) it follows that the first two entries in the one-line notation of ๐‘ค are also strictly greater than 1, and 2โˆˆ๐ท๐‘…(๐‘ค). 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 ๐‘ค<๐‘ค๎…ž, then, since 2โˆˆ๐ท๐‘…(๐‘ค) in both cases under consideration, we must have that one of the first two entries of ๐‘ค is equal to 1, but we have just seen that is impossible. Hence ๐‘คฬธ<๐‘คโ€ฒ. The assertion that ฬธ๐‘Ÿ๐‘œโ„“โ„“(๐‘ค)<๐‘คโ€ฒ 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 ๐‘ค, ๐‘คโ€ฒโˆˆHess(โ„Ž)๐‘†1. Suppose one of the following conditions hold:(i)๐‘ค and ๐‘คโ€ฒ are of the same type, or(ii)๐‘ค is of Peterson type and does not contain 321, and ๐‘คโ€ฒ is either 312-type or 231-type, or(iii)๐‘ค is either 312-type or 231-type, and ๐‘คโ€ฒ is of Peterson type.Then, ๐‘ค<๐‘ค๎…ži๏ฌ€๐’œ๎€ท๐‘ค(๐‘ค)โŠ†๐’œ๎…ž๎€ธ.(4.35)

Proof. Since the lemma above shows that ๐‘ค<๐‘คโ€ฒ implies ๐’œ(๐‘ค)โŠ†๐’œ(๐‘คโ€ฒ), for all cases it suffices to show the reverse implication. First suppose ๐‘ค and ๐‘คโ€ฒ are of the same type and ๐’œ(๐‘ค)โŠ†๐’œ(๐‘คโ€ฒ). 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 ๐‘ค<๐‘คโ€ฒ. Now suppose ๐‘ค is of Peterson type and does not contain 321 and ๐‘คโ€ฒ is either of 312 type or 231-type. Then since ฬธ{1,2}โŠ†๐’œ(๐‘ค), either 1โˆ‰๐’œ(๐‘ค) or 2โˆ‰๐’œ(๐‘ค). From the explicit reduced word decompositions of 312 or 231-type fixed points chosen above it can be seen that ๐‘คโ€ฒ is Bruhat-greater than both ๐‘ค๐’œ(๐‘ค๎…ž)โงต{1} and ๐‘ค๐’œ(๐‘ค๎…ž)โงต{2}. The claim now follows from Lemma 4.23. Finally suppose ๐‘ค is either 312-type or 231-type and ๐‘คโ€ฒ is of Peterson type. Since ๐’œ(๐‘ค)โŠ†๐’œ(๐‘คโ€ฒ) we know from Lemma 4.23 that ๐‘ค๐’œ(๐‘ค)<๐‘ค๐’œ(๐‘ค๎…ž)=๐‘คโ€ฒ. Lemma 4.24 shows that ๐‘ค<๐‘ค๐’œ(๐‘ค) 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 ๐’œโŠ†{1,2,โ€ฆ,๐‘›โˆ’1}. Recall we denote by ๐‘ค๐’œ the Peterson-type fixed point associated to ๐’œ. For the purposes of this discussion we let ๐‘ข๐’œ (resp., ๐‘ฃ๐’œ) denote the 312-type (resp., 231-type) fixed point with associated subset ๐’œ. Thus for ๐’œ=[1,๐‘Ž] for some ๐‘Ž with 2โ‰ค๐‘Žโ‰ค๐‘›โˆ’1, we have๐‘ข[1,๐‘Ž]=(๐‘Ž+1๐‘Žโ‹ฏ312๐‘Ž+2๐‘Ž+3โ‹ฏ๐‘›)โˆ’1=๐‘Ž๐‘Ž+1๐‘Žโˆ’1๐‘Žโˆ’2โ‹ฏ21๐‘Ž+2๐‘Ž+3โ‹ฏ๐‘›,(4.36)๐‘ฃ[1,๐‘Ž]=(2๐‘Ž+1๐‘Žโ‹ฏ431๐‘Ž+2๐‘Ž+3โ‹ฏ๐‘›)โˆ’1=๐‘Ž+11๐‘Ž๐‘Žโˆ’1โ‹ฏ32๐‘Ž+2๐‘Ž+3โ‹ฏ๐‘›(4.37) in one-line notation. For general subsets๎€บ๐‘Ž๐’œ=1,๐‘Ž2๎€ปโˆช๎€บ๐‘Ž3,๐‘Ž4๎€ป๎€บ๐‘Žโˆชโ‹ฏโˆช๐‘šโˆ’1,๐‘Ž๐‘š๎€ป,(4.38) with ๐‘Ž1=1 and ๐‘Ž2โ‰ฅ2, the definitions 312-type and 231-type fixed points imply that๐‘ข๐’œ=๐‘ข[๐‘Ž1,๐‘Ž2]๐‘ค[๐‘Ž3,๐‘Ž4]โ‹ฏ๐‘ค[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š],(4.39)๐‘ฃ๐’œ=๐‘ฃ[๐‘Ž1,๐‘Ž2]๐‘ค[๐‘Ž3,๐‘Ž4]โ‹ฏ๐‘ค[๐‘Ž๐‘šโˆ’1,๐‘Ž๐‘š].(4.40)

Lemma 4.29. Let ๐’œ, โ„ฌ be subsets of {1,2,โ€ฆ,๐‘›โˆ’1} and let ๎€บ๐‘Ž๐’œ=1,๐‘Ž2๎€ปโˆช๎€บ๐‘Ž3,๐‘Ž4๎€ป๎€บ๐‘Žโˆชโ‹ฏโˆช๐‘šโˆ’1,๐‘Ž๐‘š๎€ป๎€บ๐‘,โ„ฌ=1,๐‘2๎€ปโˆช๎€บ๐‘3,๐‘4๎€ป๎€บ๐‘โˆชโ‹ฏโˆช๐‘šโˆ’1,๐‘๐‘š๎€ป(4.41) be the respective decompositions into maximal consecutive substrings. Assume both ๐’œ and โ„ฌ contain {1,2}. Let ๐‘ค๐’œ (resp., ๐‘ฃ๐’œ) be the Peterson-type (resp., 231-type) fixed point corresponding to ๐’œ and let ๐‘ขโ„ฌ be the 312-type fixed point corresponding to โ„ฌ. Then ๐‘ค๐’œ<๐‘ขโ„ฌ๎€ทresp.,๐‘ฃ๐’œ<๐‘ขโ„ฌ๎€ธi๏ฌ€๐’œโŠ†โ„ฌ,๐‘2โ‰ฅ๐‘Ž2+1.(4.42)

Proof. We begin by recalling two basic observations about Bruhat order in ๐‘†๐‘›. Both follow straightforwardly from its definition in terms of reduced word decompositions. Suppose ๐‘ค,๐‘คโ€ฒโˆˆ๐‘†๐‘› and assume that