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

Abstract

We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form h = (h(1), n . . . , 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 S n on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the S n -invariant subring of the cohomology ring.

Manifolds of Isospectral Matrices and Hessenberg Varieties

“…It is known (see [16] or [1] and references therein), that Y h is a smooth variety of complex dimension d, its odd cohomology vanish, and Y h is a GKM-manifold with respect to the action of a compact torus T n Ă pCˆq n . HT pX h q -HT pY h q.…”

“…Its vertices are all permutations σ P E, and two vertices σ and τ are incident if τ " σpi, jq and either i ă j ď hpiq or j ă i ď hpjq. The unlabeled GKM-graph of Y h is the same, see [1].…”

“…(3) We recall the result of [1] on the structure of equivariant cohomology ring of a semisimple Hessenberg variety:…”

“…Now, since HT pX h q is isomorphic to HT pY h q as ring, we see that H˚pX h q is generated in degree 2 if and only if H˚pY h q is generated in degree 2. It was shown in [1] that the Hessenberg function h " php1q, n, . .…”

Crystallization of poly(3‐hydroxybutyrate‐ co ‐3‐hydroxyhexanoate) during melt extrusion promoted by residual crystals

“…Various [12] Although talc, micronized mica, boron nitride, calcium carbonate, organic phosphonic or phosphinic acids and their esters or derivatives, metal oxides (groups I-V of the periodic table), metal hydroxides, and saturated/unsaturated carboxylic acid salts have been tested as nucleating agents for PHAs, [13] their performance has been unsatisfactory. As other low molecular weight additives effective as crystal nucleating agents, amide compounds such as oxalamide derivatives, [14][15][16] behenic acid amide, [17] and polyhydric alcohol compounds such as pentaerythritol [18] have been studied.…”