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

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.