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A survey of recent developments on Hessenberg varieties
H Abe, T Horiguchi - Schubert Calculus and Its Applications in …, 2020 - Springer
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 …
rich connections of their cohomology and combinatorics. In particular, we will see how …
The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
H Abe, M Harada, T Horiguchi… - International …, 2019 - academic.oup.com
Let be a fixed positive integer and a Hessenberg function. The main results of this paper are
two-fold. First, we give a systematic method, depending in a simple manner on the …
two-fold. First, we give a systematic method, depending in a simple manner on the …
The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
We compute the expansion of the cohomology class of the permutahedral variety in the
basis of Schubert classes. The resulting structure constants are expressed as a sum of …
basis of Schubert classes. The resulting structure constants are expressed as a sum of …
Geometry of regular Hessenberg varieties
H Abe, N Fujita, H Zeng - Transformation Groups, 2020 - Springer
Let gg be a complex semisimple Lie algebra. For a regular element x in gg and a
Hessenberg space H⊆ gg, we consider a regular Hessenberg variety X (x, H) in the ag …
Hessenberg space H⊆ gg, we consider a regular Hessenberg variety X (x, H) in the ag …
Mixed Eulerian numbers and Peterson Schubert calculus
T Horiguchi - International Mathematics Research Notices, 2024 - academic.oup.com
Let be a root system. Postnikov introduced and studied the mixed-Eulerian numbers. These
numbers indicate the mixed volumes of-hypersimplices. As specializations of these …
numbers indicate the mixed volumes of-hypersimplices. As specializations of these …
Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
In this paper, we study the geometry of various Hessenberg varieties in type A, as well as
families thereof. Our main results are as follows. We find explicit and computationally …
families thereof. Our main results are as follows. We find explicit and computationally …
The Betti numbers of regular Hessenberg varieties are palindromic
M Precup - Transformation Groups, 2018 - Springer
Abstract Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs
describing a representation of the symmetric group on the cohomology of regular …
describing a representation of the symmetric group on the cohomology of regular …
Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
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 …
varieties in flag manifolds using the notion of pattern avoidance. This was the first time …
Toric orbifolds associated with partitioned weight polytopes in classical types
T Horiguchi, M Masuda, J Shareshian, J Song - Selecta Mathematica, 2024 - Springer
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∈ E not orthogonal to any of the roots in Φ, we consider …
associated Weyl group. For a point p∈ E not orthogonal to any of the roots in Φ, we consider …
The cohomology rings of regular semisimple Hessenberg varieties for
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 …
Hessenberg functions are of the form $ h=(h (1), n\dots, n) $ in Lie type $ A_ {n-1} $. The …