scholarly journals The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials

2018 ◽  
Vol 94 (9) ◽  
pp. 87-92 ◽  
Author(s):  
Tatsuya Horiguchi
Keyword(s):  
Cohomology Rings ◽  
Schubert Polynomials ◽  
Hessenberg Varieties

scholarly journals A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

2020 ◽  
Author(s):  
Megumi Harada ◽  
Tatsuya Horiguchi ◽  
Satoshi Murai ◽  
Martha Precup ◽  
Julianna Tymoczko
Keyword(s):  
Cohomology Rings ◽  
Hessenberg Varieties

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, 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 semisimple Hessenberg varieties for $h = (h(1),n,\dotsc,n)$

2019 ◽  
Vol 10 (1) ◽  
pp. 27-59
Author(s):  
Hiraku Abe ◽  
Tatsuya Horiguchi ◽  
Mikiya Masuda
Keyword(s):  
Cohomology Rings ◽  
Hessenberg Varieties

scholarly journals Schubert polynomials and classes of Hessenberg varieties

2010 ◽  
Vol 323 (10) ◽  
pp. 2605-2623 ◽  
Author(s):  
Dave Anderson ◽  
Julianna Tymoczko
Keyword(s):  
Schubert Polynomials ◽  
Hessenberg Varieties

scholarly journals Manifolds of Isospectral Matrices and Hessenberg Varieties

2020 ◽  
Author(s):  
Anton Ayzenberg ◽  
Victor Buchstaber
Keyword(s):  
Smooth Manifold ◽  
Flag Variety ◽  
Natural Action ◽  
Cohomology Rings ◽  
Compact Torus ◽  
Hessenberg Varieties ◽  
Toda Flow ◽  
Odd Degree ◽  
Complete Flag ◽  
The Given

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.

scholarly journals Upper bounds of Schubert polynomials

2021 ◽  
Author(s):  
Neil Jiuyu Fan ◽  
Peter Long Guo
Keyword(s):  
Upper Bounds ◽  
Schubert Polynomials

scholarly journals On the Multiplication of Schubert Polynomials

1998 ◽  
Vol 20 (1) ◽  
pp. 73-97 ◽  
Author(s):  
Rudolf Winkel
Keyword(s):  
Schubert Polynomials
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