Poset Pinball, the Dimension Pair Algorithm, and Type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> Regular Nilpotent Hessenberg Varieties

Bayegan, Darius; Harada, Megumi

doi:10.5402/2012/254235

Cited by 6 publications

(23 citation statements)

References 21 publications

(111 reference statements)

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“…A similar analysis by Bayegan and the second author in a special case of regular nilpotent Hessenberg varieties [2] yields a poset-uppertriangular basis in the sense of [9]. In contrast to the results in [2], in the present manuscript we find that the module basis is not poset-upper-triangular; this is the first such example in the literature. In addition, a straightforward consequence of our proof is that a simple change of variables yields a module basis which is not a poset pinball basis but is poset-upper-triangular.…”

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confidence: 54%

“…Open questions 24 References 24 1 We use English notation for Young diagrams. 2 We work with cohomology with coefficients in C throughout, and hence omit it from our notation.POSET PINBALL, HIGHEST FORMS, AND (n − 2, 2) SPRINGER VARIETIES 3 between permissible fillings and S 1 -fixed points of the Springer variety. The module basis is obtained by taking images under the natural projection map H * T (Fℓags(C n )) → H * S 1 (S (n−2,2) ), to be described in detail below, of a subset of the T -equivariant Schubert classes in H * T (Fℓags(C n )).…”

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confidence: 99%

“…Section 4 contains our results on the bijection between permissible fillings of a Young diagram and the S 1 -fixed points of Hessenberg varieties. Section 5 is a mainly expository section which recalls the terminology and definitions of poset pinball and the dimension pair algorithm in [2,9]. In Sections 6 and 7, poset pinball for the case of (n − 2, 2) Springer varieties is studied in detail.…”

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Poset Pinball, Highest Forms, and $(n-2,2)$ Springer Varieties

Dewitt¹,

Harada²

2012

Electron. J. Combin.

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In this manuscript we study type A nilpotent Hessenberg varieties equipped with a natural S 1action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition λ = (n − 2, 2) for n ≥ 4. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with n boxes with the alphabet {1, 2, . . . , n}. Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymoczko. Second, for a nilpotent operator N and Hessenberg function h, we construct an explicit bijection between the S 1 -fixed points of the nilpotent Hessenberg variety Hess(N, h) and the set of (h, λ N )-permissible fillings of the Young diagram λ N . Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the S 1equivariant cohomology of type A Springer varieties S (n−2,2) associated to Young diagrams of shape (n − 2, 2) for n ≥ 4. Specifically, we use the dimension pair algorithm for Betti-acceptable pinball described by Bayegan and Harada to specify a subset of the equivariant Schubert classes in the T -equivariant cohomology of the flag variety Fℓags(C n ) ∼ = GL(n, C)/B which maps to a module basis of H * S 1 (S (n−2,2) ) under the projection map H * T (Fℓags(C n )) → H * S 1 (S (n−2,2) ). Our poset pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A straightforward consequence of our proof is that there exists a simple and explicit change of basis which transforms our poset pinball basis to a poset-upper-triangular module basis for H * S 1 (S (n−2,2) ).We close with open questions for future work. CONTENTS 1. Introduction 1 2. Nilpotent Hessenberg varieties and S 1 -actions 3 3. Adjacent-pair matrices and highest forms of nilpotent operators 4 4. S 1 -fixed points in Hessenberg varieties and permissible fillings 12 5. Betti-acceptable pinball and linear independence 14 6. Small-n cases: n = 4 and n = 5 17 7. A poset pinball module basis for (n − 2, 2) Springer varieties 18 8. Open questions 24 References 24 1 We use English notation for Young diagrams. 2 We work with cohomology with coefficients in C throughout, and hence omit it from our notation.POSET PINBALL, HIGHEST FORMS, AND (n − 2, 2) SPRINGER VARIETIES 3 between permissible fillings and S 1 -fixed points of the Springer variety. The module basis is obtained by taking images under the natural projection map H * T (Fℓags(C n )) → H * S 1 (S (n−2,2) ), to be described in detail below, of a subset of the T -equivariant Schubert classes in H * T (Fℓags(C n )). A similar analysis by Bayegan and the second author in a special case of regular nilpotent Hessenberg varieties [2] yields a poset-uppertriangular basis in the sense of [9]. In contrast to the results in [2], in the present manuscript we find that the module basis is not poset-upper-triangular; this is t...

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confidence: 99%

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Poset Pinball, Highest Forms, and $(n-2,2)$ Springer Varieties

Dewitt¹,

Harada²

2012

Electron. J. Combin.

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“…Using the association of the polynomials f i,j with the (i, j)-th entry of the matrix (3.5), the ideal I h can visually be described as being generated by the f i,j in the boxes at the bottom of each column in the the picture associated to the Hessenberg subspace H(h) defined in (2.2) (see Figure 1). For instance, when h = (3,3,4,5,6,6), the generators are f 3,1 , f 3,2 , f 4,3 , f 5,4 , f 6,5 , f 6,6 .…”

Section: Statement Of Theorem 33 the Equivariant Version Of Theorem Amentioning

confidence: 99%

“…More precisely, we say that a Hessenberg function meets the ℓ-th lower diagonal if there exists some j ∈ [n] such that h(j) − j ≥ ℓ. For example, for h = (2, 3, 4, 5, 5) a Peterson Hessenberg function, the Hessenberg subspace meets the 0-th and 1st lower diagonals, whereas for h = (3,4,4,5,5), the Hessenberg subspace also meets the 2nd lower diagonal. The lowest lower diagonal which h meets is evidently max j∈[n] {h(j) − j}.…”

Section: Properties Of the F Ijmentioning

confidence: 99%

The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Abe

Harada

Horiguchi

et al. 2017

International Mathematics Research Notices

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137

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Abstract. Let n be a fixed positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function h, for producing an explicit presentation by generators and relations of the cohomology ring H * (Hess(N, h)) with Q coefficients of the corresponding regular nilpotent Hessenberg variety Hess(N, h). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring H * (Hess(N, h)) of the regular nilpotent Hessenberg variety and the Sn-invariant subring H * (Hess(S, h)) Sn of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the Sn-action on H * (Hess(S, h)) defined by Tymoczko). Our second main result implies that dim Q H k (Hess(N, h)) = dim Q H k (Hess(S, h)) Sn for all k and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.

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Equivariant cohomology and local invariants of Hessenberg varieties

Insko¹

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Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety.In this thesis I give a geometric proof that the cohomology of the flag variety surjects onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the singular loci of a large family of regular nilpotent Hessenberg varieties;and I describe the equivariant cohomology of any regular nilpotent Hessenberg variety whose cohomology is generated by its degree two classes. Abstract Approved:Thesis Supervisor ACKNOWLEDGEMENTSThank you to everyone who helped me along my way. I appreciate my family for their continuing support and encouragement and my partner Elizabeth for all her love and patience.

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

Cited by 6 publications

References 21 publications

Poset Pinball, Highest Forms, and $(n-2,2)$ Springer Varieties

Poset Pinball, Highest Forms, and $(n-2,2)$ Springer Varieties

The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Equivariant cohomology and local invariants of Hessenberg varieties

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