The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A

By Hiraku Abe, Megumi Harada, T. Horiguchi, M. Masuda. 2015

62 citations recorded by the source corpus.

Let $n$ 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 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 $H^\ast(Hess(\mathsf{N},h))$ with $\mathbb{Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety $Hess(\mathsf{N},h)$. 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 $H^*(Hess(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $S_n$-invariant subring $H^*(Hess(\mathsf{S},h))^{S_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $S_n$-action on $H^*(Hess(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{N},h)) = \mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{S},h))^{S_n}$ 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, 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a1bc431b49f63fbdcc8c3a97569f3d45948c2cfb

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Citations in the corpus

The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, background): “kablerelationshipbetweentheBettinumbersofdifferentHessenberg varieties. (Directcomputationsofcohomology ringsofcertainHessenbergvarietiesalsoyieldpartialproofs of the Shareshian-Wachs conjecture; see [1,2].) It then follows that, in order to prove the graded StanleyStembridgeconjecture, it sufﬁcesto”
Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “…announcement [2], which explains how to compute the multiplicity of the trivial representation in terms of the regular nilpotent Hessenberg variety, partially inspired our own proof Full details of the work of AHHM appeared on the arXiv in [1] shortly after the ﬁrst draft [7] of this paper.”
Chromatic quasisymmetric functions (2014, background): “ of the left hand side of (10.4) that is identical to the right hand side of (6.6), thereby providing an independent proof of the fact that the coeﬃcients of sn on both sides of (10.4) are equal, see [AHHM]. Clearman, Hyatt, Shelton and Skandera [CHSS1, CHSS2] have given an algebraic interpretation ”