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

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-34 ◽  
Author(s):  
Darius Bayegan ◽  
Megumi Harada
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Type A ◽  
Successful Outcome ◽  
Combinatorial Game ◽  
Schubert Polynomials ◽  
Hessenberg Varieties ◽  
Hessenberg Variety ◽  
Springer Variety ◽  
Special Case

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 HS1*(pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.

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

2017 ◽  
Vol 2019 (17) ◽  
pp. 5316-5388 ◽  
Author(s):  
Hiraku Abe ◽  
Megumi Harada ◽  
Tatsuya Horiguchi ◽  
Mikiya Masuda
Keyword(s):  
Commutative Algebra ◽  
Cohomology Ring ◽  
Regular Sequence ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Hessenberg Varieties ◽  
Special Cases ◽  
Hessenberg Variety ◽  
Fixed Positive Integer ◽  
Special Case

AbstractLet $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$ a Hessenberg function. The main results of this paper are two-fold. 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 $H^\ast({\mathrm{Hess}}(\mathsf{N},h))$ with ${\mathbb Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety ${\mathrm{Hess}}(\mathsf{N},h)$. 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 $H^*({\mathrm{Hess}}(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $\mathfrak{S}_n$-invariant subring $H^*({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $\mathfrak{S}_n$-action on $H^*({\mathrm{Hess}}(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{N},h)) = \mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ 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, and independently by Guay–Paquet, but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [2].

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

10.37236/2126 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Barry Dewitt ◽  
Megumi Harada
Keyword(s):  
Young Diagram ◽  
Equivariant Cohomology ◽  
Type A ◽  
Linear Operators ◽  
Young Diagrams ◽  
Adjacent Pair ◽  
Open Questions ◽  
Hessenberg Varieties ◽  
Study Type ◽  
Future Work

In this manuscript we study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-action 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 $\lambda= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. 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 $(h,\lambda_N)$-permissible fillings of the Young diagram $\lambda_N$. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the $S^1$-equivariant cohomology of type $A$ Springer varieties $\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for $n\geq 4$. 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 $\mathbb{T}$-equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\mathbb{C}^n) \cong GL(n,\mathbb{C})/B$ which maps to a module basis of $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection map $H^*_\mathbb{T}(\mathcal{F}\ell ags(\mathbb{C}^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. 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 $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with open questions for future work.

scholarly journals Hessenberg varieties and hyperplane arrangements

2020 ◽  
Vol 2020 (764) ◽  
pp. 241-286 ◽  
Author(s):  
Takuro Abe ◽  
Tatsuya Horiguchi ◽  
Mikiya Masuda ◽  
Satoshi Murai ◽  
Takashi Sato
Keyword(s):  
Cohomology Ring ◽  
Linear Algebraic Group ◽  
Affirmative Answer ◽  
Hyperplane Arrangements ◽  
Hessenberg Varieties ◽  
Hessenberg Variety ◽  
Positive Roots ◽  
Lefschetz Property ◽  
Complex Linear ◽  
Semisimple Complex

AbstractGiven a semisimple complex linear algebraic group {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement {\mathcal{A}_{I}}, the regular nilpotent Hessenberg variety {\operatorname{Hess}(N,I)}, and the regular semisimple Hessenberg variety {\operatorname{Hess}(S,I)}. We show that a certain graded ring derived from the logarithmic derivation module of {\mathcal{A}_{I}} is isomorphic to {H^{*}(\operatorname{Hess}(N,I))} and {H^{*}(\operatorname{Hess}(S,I))^{W}}, the invariants in {H^{*}(\operatorname{Hess}(S,I))} under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety {G/B}.This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map {H^{*}(G/B)\to H^{*}(\operatorname{Hess}(N,I))} announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for {\operatorname{Hess}(N,I)}, despite the fact that it is a singular variety in general.

scholarly journals A Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties

10.37236/425 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Aba Mbirika
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Group Action ◽  
Cohomology Ring ◽  
Geometric Construction ◽  
Nilpotent Operator ◽  
Hessenberg Varieties ◽  
New Evidence ◽  
Springer Variety ◽  
Two Parameter

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.

scholarly journals Springer’s Weyl Group Representation via Localization

2017 ◽  
Vol 60 (3) ◽  
pp. 478-483 ◽  
Author(s):  
Jim Carrell ◽  
Kiumars Kaveh
Keyword(s):  
Weyl Group ◽  
Equivariant Cohomology ◽  
Group Representation ◽  
General Theorem ◽  
Torus Action ◽  
Cohomology Algebra ◽  
Reductive Algebraic Group ◽  
Point Set ◽  
Closed Subvariety ◽  
Springer Variety

AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

A new equivariant cohomology ring

2019 ◽  
Vol 295 (3-4) ◽  
pp. 1163-1182 ◽  
Author(s):  
Bohui Chen ◽  
Cheng-Yong Du ◽  
Tiyao Li
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring

scholarly journals OCNEANU'S TRACE AND STARKEY'S RULE

2003 ◽  
Vol 12 (07) ◽  
pp. 899-904 ◽  
Author(s):  
MEINOLF GECK ◽  
NICOLAS JACON
Keyword(s):  
Simple Proof ◽  
Hecke Algebras ◽  
Jones Polynomial ◽  
Main Tool ◽  
Type A ◽  
Character Tables ◽  
Special Case ◽  
Knots And Links

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

scholarly journals Double theta polynomials and equivariant Giambelli formulas

2015 ◽  
Vol 160 (2) ◽  
pp. 353-377 ◽  
Author(s):  
HARRY TAMVAKIS ◽  
ELIZABETH WILSON
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Generators And Relations ◽  
Raising Operators ◽  
Schubert Classes ◽  
Symplectic Grassmannians

AbstractWe use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

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