Journal of Combinatorics

Volume 10 (2019)

Number 1

The cohomology rings of regular semisimple Hessenberg varieties for $h = (h(1),n,\dotsc,n)$

Pages: 27 – 59

DOI: https://dx.doi.org/10.4310/JOC.2019.v10.n1.a2

Authors

Hiraku Abe (Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi-ku, Osaka, Japan)

Tatsuya Horiguchi (Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan; and Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi-ku, Osaka, Japan)

Mikiya Masuda (Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka, Japan)

Abstract

We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h = (h(1),n,\dotsc,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel’s presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $\mathfrak{S}_n$-invariant subring of the cohomology ring.

Keywords

Hessenberg varieties, flag varieties, cohomology rings, representations of symmetric groups, Shareshian-Wachs conjecture, Stanley-Stembridge conjecture

Full Text (PDF format)

Hako Abe’s research partially supported by a JSPS Grant-in-Aid for Young Scientists (B): 15K17544 and a JSPS Research Fellowship for Young Scientists Postdoctoral Fellow 16J04761.

Tatsuya Horiguchi’s research partially supported by JSPS Grant-in-Aid for JSPS Fellows 15J09343.

Mikiya Masuda’s research partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05152.

Received 12 April 2017

Published 7 December 2018