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 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 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 The connectedness of Hessenberg varieties

2015 ◽  
Vol 437 ◽  
pp. 34-43 ◽  
Author(s):  
Martha Precup
Keyword(s):  
Hessenberg Varieties

scholarly journals Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

2018 ◽  
Vol 24 (3) ◽  
pp. 2129-2163 ◽  
Author(s):  
Hiraku Abe ◽  
Lauren DeDieu ◽  
Federico Galetto ◽  
Megumi Harada
Keyword(s):  
Hessenberg Varieties

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals Inversions Within Restricted Fillings of Young Tableaux

10.37236/1030 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Sarah Iveson
Keyword(s):  
Upper Bound ◽  
Exact Expression ◽  
Flag Varieties ◽  
Young Tableaux ◽  
Geometric Properties ◽  
Number Of Components ◽  
Hessenberg Varieties ◽  
Special Cases

In this paper we study inversions within restricted fillings of Young tableaux. These restricted fillings are of interest because they describe geometric properties of certain subvarieties, called Hessenberg varieties, of flag varieties. We give answers and partial answers to some conjectures posed by Tymoczko. In particular, we find the number of components of these varieties, give an upper bound on the dimensions of the varieties, and give an exact expression for the dimension in some special cases. The proofs given are all combinatorial.

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