The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement

Abstract

Let $n$ be a fixed positive integer and $h:\{1,2,...,n\}\to \{1,2,...,n\}$ a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with $\mathbb{Q}$ coefficients) of any regular nilpotent Hessenberg variety $\mathrm{Hess}(h)$ in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function $h$, which produces the $n$ defining relations $\{f_{h(j),j}{\}}_{j=1}^n$ in the equivariant cohomology ring. Our result generalizes known results: for the case $h=(2,3,4,...,n,n)$, which corresponds to the Peterson variety ${\mathrm{Pet}}_n$, we recover the presentation of $H_S^{\ast }({\mathrm{Pet}}_n)$ given previously by Fukukawa, Harada, and Masuda. Moreover, in the case $h=(n,n,...,n)$, for which the corresponding regular nilpotent Hessenberg variety is the full flag variety $\mathrm{Flags}({\mathbb{C}}^n)$, we can explicitly relate the generators of our ideal with those in the usual Borel presentation of the cohomology ring of $\mathrm{Flags}({\mathbb{C}}^n)$. The proof of our main theorem includes an argument that the restriction homomorphism $H_T^{\ast }(\mathrm{Flags}({\mathbb{C}}^n))\to H_S^{\ast }(\mathrm{Hess}(h))$ is surjective. In this research announcement, we briefly recount the context and state our results; we also give a sketch of our proofs and conclude with a brief discussion of open questions. A manuscript containing more details and full proofs is forthcoming.

8 pages. Expository changes based on comments by a referee

References in corpus (2)

• Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
• The torus equivariant cohomology rings of Springer varieties