arXiv subjects

Tatsuya Horiguchi

Publications and source records attributed to Tatsuya Horiguchi.

At least 19 recordsLinked to original sources

arXiv2025

Coordinate rings of regular semisimple Hessenberg varieties and cohomology rings of regular nilpotent Hessenberg varieties

The polynomials $f_{i,j}$ are introduced by Abe-Harada-Horiguchi-Masuda to produce an explicit presentation by generators and relations of the cohomology rings of regular nilpotent Hessenberg varieties. In this paper we quantize the polynomials $f_{i,j}$ by a method of Fomin-Gelfand-Postnikov. Our main result states that their quantizations $F_{i,j}$ are related to the coordinate rings of regular semisimple Hessenberg varieties. This result yields a connection between the coordinate rings of regular semisimple Hessenberg varieties and the cohomology rings of regular nilpotent Hessenberg varieties. We also provide the quantized recursive formula for $F_{i,j}$.

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arXiv2025

Notes on the cohomology of partial Hessenberg varieties

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

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arXiv2025

Regular nilpotent partial Hessenberg varieties

Let $G$ be a complex semisimple linear algebraic group. Fix a subset $Θ$ of simple roots. Given a lower ideal $I$ in positive roots, one can define the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$ in the full flag variety $G/B$. For a $Θ$-ideal $I$ (which is a special lower ideal), we can define the regular nilpotent partial Hessenberg variety $\mbox{Hess}_Θ(N,I)$ in the partial flag variety $G/P$. In this manuscript we first provide a summand formula and a product formula for the Poincaré polynomial of regular nilpotent partial Hessenberg varieties. It is a well-known result from Bernstein-Gelfand-Gelfand that the cohomology ring of the partial flag variety $G/P$ is isomorphic to the invariants in the cohomology ring of the full flag variety $G/B$ under an action of the parabolic Weyl group $W_Θ$ generated by $Θ$. We generalize this result to regular nilpotent partial Hessenberg varieties. More concretely, we give an isomorphism between the cohomology ring of a regular nilpotent partial Hessenberg variety $\mbox{Hess}_Θ(N,I)$ and the $W_Θ$-invariant subring of the cohomology ring of the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$. Furthermore, we provide a description of the cohomology ring for a regular nilpotent partial Hessenberg variety $\mbox{Hess}_Θ(N,I)$ in terms of the $W_Θ$-invariants in the logarithmic derivation module of the ideal arrangement $\mathcal{A}_I$, which is a generalization of the result by Abe-Masuda-Murai-Sato with the author.

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arXiv2025

Uniform bases for ideal arrangements

In this paper we introduce and study uniform bases for the ideal arrangements in all Lie types. Explicit uniform bases are given by Abe-Horiguchi-Masuda-Murai-Sato for types $A,B,C,G$ and we provide them for other types. Combining the explicit uniform bases with the work of Abe-Horiguchi-Masuda-Murai-Sato, we also obtain explicit presentations of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types.

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arXiv2025

Gamma vectors of partitioned permutohedra

We determine that $γ$-vectors of partitioned permutohedra, thereby generalizing a result of Foata and Schützenberger. Our result is closely related to a result of Athanasiadis on the representation of the symmetric group on the cohomology of the permutohedral variety. We explain how to derive Athanasiadis' result from ours and vice versa.

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arXiv2024

Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\mbox{GL}_n(\mathbb{C})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\mbox{Pet}_n$ and the opposite Schubert cell associated with the identity element $Ω_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\mbox{GL}_n(\mathbb{C})/B$ is given by Ciocan-Fontanine and Givental-Kim. In this paper we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\mbox{Hess}(N,h)$ and $Ω_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. In other words, we generalize the Peterson's statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan-Fontanine and Givental-Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\mbox{Hess}(N,h_m)$ and $Ω_e^\circ$ is the intersection of certain Schubert variety and $Ω_e^\circ$ where $h_m=(m,n,\ldots,n)$ for $1<m<n$. We also see that $\mbox{Hess}(N,h_2) \cap Ω_e^\circ$ is related with the cyclic quotient singularity.

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arXiv2024

The cohomology rings of regular nilpotent Hessenberg varieties

This manuscript is a contributed chapter in the forthcoming CRC Press volume, titled the Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety. The book, as a whole, is aimed at a diverse audience of researchers and graduate students seeking an expository introduction to the area. In our chapter, we give an overview of some of the past research on the cohomology rings of regular nilpotent Hessenberg varieties, with no claim to being exhaustive. For the purposes of this manuscript, we focus mainly on the case of Lie type A, with some brief remarks on the general Lie types. We end the chapter with a selection of topics currently active in this area.

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arXiv2024

Toward cohomology rings of intersections of Peterson varieties and Richardson varieties

Peterson varieties are subvarieties of flag varieties and their (equivariant) cohomology rings are given by Fukukawa-Harada-Masuda in type A and soon later the author with Harada and Masuda gives an explicit presentation of the (equivariant) cohomology rings of Peterson varieties for arbitrary Lie types. In this note we study the (equivariant) cohomology ring of the intersections of Peterson variety with Schubert, opposite Schubert, and Richardson varieties in more general. By the work of Goldin-Mihalcea-Singh, the intersections of Peterson variety with Schubert varieties are naturally identified with smaller Peterson varieties, so the problem reduces to the problem for opposite Schubert intersections. In this note we provide a technical statement for (equivariant) cohomology ring of a subvariety with some conditions of Peterson variety. By using the statement, we calculate the (equivariant) cohomology rings for some intersections of Peterson varieties with opposite Schubert varieties in type A. We also explicitly present the (equivariant) cohomology rings for some intersections of Peterson varieties with Richardson varieties in type A.

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arXiv2023

Modular law through GKM theory

The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with the dot action. A similar result holds between unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties. A recent result by Abreu-Nigro enabled us to prove these results by showing the modular law for the geometrical objects, and this is indeed done by Precup-Sommers and Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular law through GKM theory.

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arXiv2023

Geometry of Peterson Schubert calculus in type A and left-right diagrams

We introduce an additive basis of the integral cohomology ring of the Peterson variety which reflects the geometry of certain subvarieties of the Peterson variety. We explain the positivity of the structure constants from a geometric viewpoint, and provide a manifestly positive combinatorial formula for them. We also prove that our basis coincides with the additive basis introduced by Harada-Tymoczko.

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arXiv2023

Mixed Eulerian numbers and Peterson Schubert calculus

Let $Φ$ be a root system. Postnikov introduced and studied the mixed $Φ$-Eulerian numbers. These numbers indicate the mixed volumes of $Φ$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget-Spink-Tseng gave a simple computation for the mixed $Φ$-Eulerian numbers when $Φ$ is of type $A$. In this paper we connect a relation between mixed $Φ$-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget-Spink-Tseng in terms of left-right diagrams which were introduced by Abe-Horiguchi-Kuwata-Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed $Φ$-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.

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arXiv2023

An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties

In this paper we construct an additive basis for the cohomology ring of a regular nilpotent Hessenberg variety which is obtained by extending all Poincaré duals of smaller regular nilpotent Hessenberg subvarieties. In particular, all of the Poincaré duals of smaller regular nilpotent Hessenberg subvarieties are linearly independent.

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arXiv2021

Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system $Φ$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $Φ$, we consider the $W$-permutohedron $P_W$, which is the convex hull of the $W$-orbit of $p$. The representation of $W$ on the rational cohomology ring $H^\ast(X_Φ)$ of the toric variety $X_Φ$ associated to (the normal fan to) $P_W$ has been studied by various authors. Let $\{s_1,\ldots,s_n\}$ be a complete set of simple reflections in $W$. For $K \subseteq [n]$, let $W_K$ be the standard parabolic subgroup of $W$ generated by $\{s_k:k \in K\}$. We show that the fixed subring $H^\ast(X_Φ)^{W_K}$ is isomorphic to the cohomology ring of the toric variety $X_Φ(K)$ associated to a polytope obtained by intersecting $P_W$ with half-spaces bounded by reflecting hyperplanes for the given generators of $W_K$. By a result of Balibanu--Crooks, the cohomology rings $H^\ast(X_Φ(K))$ are isomorphic with cohomology rings of certain regular Hessenberg varieties.

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arXiv2020

A survey of recent developments on Hessenberg varieties

This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley's chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments.

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arXiv2020

A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.

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arXiv2018

The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope

Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope $\mathrm{GZ}(λ)$ for $λ=(λ_1,λ_2,\ldots,λ_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand-Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the $α_i := λ_i-λ_{i+1}$. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial $(n-1)$-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in $\mathrm{Flag}(\mathbb{C}^n)$ as a sum of the cohomology classes of a certain set of Richardson varieties.

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arXiv2018

The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials

In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.

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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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