arXiv subjects

Tatsuya Horiguchi

Publications and source records attributed to Tatsuya Horiguchi.

25 records · Page 2Linked to original sources

arXiv2017

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

We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ 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.

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arXiv2016

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

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.

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arXiv2016

Hessenberg varieties and hyperplane arrangements

Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)\to H^*(\mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(\mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $\mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $\mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $\mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.

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arXiv2016

The $S^1$-equivariant cohomology rings of $(n-k,k)$ Springer varieties

The main result of this note gives an explicit presentation of the $S^1$-equivariant cohomology ring of the $(n-k,k)$ Springer variety (in type $A$) as a quotient of a polynomial ring by an ideal $I$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety.

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arXiv2014

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

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.

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arXiv2014

The equivariant cohomology rings of Peterson varieties in all Lie types

Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H^*_S(Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S \cong \mathbb{C}^* is a certain subgroup of a maximal torus T of G. Our description of the ideal J uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type A.

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arXiv2014

The torus equivariant cohomology rings of Springer varieties

The Springer variety of type $A$ associated to a nilpotent operator on $\mathbb{C}^n$ in Jordan canonical form admits a natural action of the $\ell$-dimensional torus $T^{\ell}$ where $\ell$ is the number of the Jordan blocks. We give a presentation of the $T^{\ell}$-equivariant cohomology ring of the Springer variety through an explicit construction of an action of the $n$-th symmetric group on the $T^{\ell}$-equivariant cohomology group. The $T^{\ell}$-equivariant analogue of so called Tanisaki's ideal will appear in the presentation.

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