28 citations · 90 across the 19 of their papers we have counts for
20 papers
Totally nonnegative Peterson variety and strongly dominant weight polytope
Hiraku Abe, Tao Gui, Haozhi Zeng
We study the totally nonnegative part of the Peterson variety in arbitrary Lie type and establish its connection to the strongly dominant weight polytope. In particular, we prove t…
Totally nonnegative part of the Peterson variety in Lie type A
Hiraku Abe, Haozhi Zeng
The Peterson variety (which we denote by ) is a subvariety of the flag variety, introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag vari…
Peterson varieties and toric orbifolds associated to Cartan matrices
Hiraku Abe, Haozhi Zeng
The Peterson variety is a remarkable variety introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. The rational cohomology ring of…
On singularity and normality of regular nilpotent Hessenberg varieties
Hiraku Abe, Erik Insko
Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert vari…
The integral cohomology rings of Peterson varieties in type A
Hiraku Abe, Haozhi Zeng
In this paper, we study the ring structure of the integral cohomology of the Peterson variety of type . We give two kinds of descriptions: (1) we show that it is is…
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…
Fano and weak Fano Hessenberg varieties
Hiraku Abe, Naoki Fujita, Haozhi Zeng
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a tor…
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…
Hessenberg varieties, Slodowy slices, and integrable systems
Hiraku Abe, Peter Crooks
This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant,…
Geometry of regular Hessenberg varieties
Hiraku Abe, Naoki Fujita, Haozhi Zeng
Let be a complex semisimple Lie algebra. For a regular element in and a Hessenberg space , we consider a regular Hessenbe…
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…
Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
Hiraku Abe, Lauren DeDieu, Federico Galetto +1
In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computatio…
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…
Hessenberg varieties for the minimal nilpotent orbit
Hiraku Abe, Peter Crooks
For a connected, simply-connected complex simple algebraic group , we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we comp…
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 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…
Young diagrams and intersection numbers for toric manifolds associated with Weyl chambers
Hiraku Abe
We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical ty…
Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
Hiraku Abe, Sara Billey
In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern…
Schur polynomials and Weighted Grassmannians
Hiraku Abe, Tomoo Matsumura
In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the coho…
Equivariant Cohomology of Weighted Grassmannians and Weighted Schubert Classes
Hiraku Abe, Tomoo Matsumura
In this paper, we study the T_w-equivariant cohomology of the weighted Grassmannians wGr(d,n) introduced by Corti-Reid where T_w is the n-dimensional torus that naturally acts on w…