29 citations · 103 across the 20 of their papers we have counts for
21 papers
Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters
Tatsuya Horiguchi
Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of…
Gamma vectors of partitioned permutohedra
Tatsuya Horiguchi, Mikiya Masuda, Takashi Sato +2
We determine that -vectors of partitioned permutohedra, thereby generalizing a result of Foata and Schützenberger. Our result is closely related to a result of Athanasiadis on t…
Modular law through GKM theory
Tatsuya Horiguchi, Mikiya Masuda, Takashi Sato
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 c…
The cohomology rings of regular nilpotent Hessenberg varieties
Megumi Harada, Tatsuya Horiguchi
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, a…
Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell
Tatsuya Horiguchi, Tomoaki Shirato
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 inters…
Toward cohomology rings of intersections of Peterson varieties and Richardson varieties
Tatsuya Horiguchi
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 a…
Toric orbifolds associated with partitioned weight polytopes in classical types
Tatsuya Horiguchi, Mikiya Masuda, John Shareshian +1
Given a root system of type , , , or in Euclidean space , let be the associated Weyl group. For a point not orthogonal to any of the roots…
Mixed Eulerian numbers and Peterson Schubert calculus
Tatsuya Horiguchi
Let be a root system. Postnikov introduced and studied the mixed -Eulerian numbers. These numbers indicate the mixed volumes of -hypersimplices. As specializations of the…
Geometry of Peterson Schubert calculus in type A and left-right diagrams
Hiraku Abe, Tatsuya Horiguchi, Hideya Kuwata +1
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 p…
An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties
Makoto Enokizono, Tatsuya Horiguchi, Takahiro Nagaoka +1
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…
Uniform bases for ideal arrangements
Makoto Enokizono, Tatsuya Horiguchi, Takahiro Nagaoka +1
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…
A survey of recent developments on Hessenberg varieties
Hiraku Abe, Tatsuya Horiguchi
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 hyp…
The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope
Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda +1
Regular semisimple Hessenberg varieties are subvarieties of the flag variety arising naturally in the intersection of geometry, representation theory,…
The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials
Tatsuya Horiguchi
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 cohomo…
The cohomology rings of regular semisimple Hessenberg varieties for
Hiraku Abe, Tatsuya Horiguchi, Mikiya Masuda
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form in Lie type . The main result…
Hessenberg varieties and hyperplane arrangements
Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda +2
Given a semisimple complex linear algebraic group and a lower ideal in positive roots of , three objects arise: the ideal arrangement , the regular nilpot…
The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
Hiraku Abe, Megumi Harada, Tatsuya Horiguchi +1
Let be a fixed positive integer and a Hessenberg function. The main results of this paper are twofold. First, we give a syste…
The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement
Hiraku Abe, Megumi Harada, Tatsuya Horiguchi +1
Let be a fixed positive integer and a Hessenberg function. The main result of this manuscript is to give a systematic method for pr…
The equivariant cohomology rings of Peterson varieties in all Lie types
Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda
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 presen…
The torus equivariant cohomology rings of Springer varieties
Hiraku Abe, Tatsuya Horiguchi
The Springer variety of type associated to a nilpotent operator on in Jordan canonical form admits a natural action of the -dimensional torus wh…
The -equivariant cohomology rings of Springer varieties
Tatsuya Horiguchi
The main result of this note gives an explicit presentation of the -equivariant cohomology ring of the Springer variety (in type ) as a quotient of a polynomial r…