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

Let $n$ be a fixed positive integer and $h: \{1,2,...,n\} \rightarrow
\{1,2,...,n\}$ 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 $H^*_S(\mathrm{Pet}_n)$ 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 $H^*_T(\mathrm{Flags}(\mathbb{C}^n))
\to H^*_S(\mathrm{Hess}(h))$ 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.