Divisors on regular Hessenberg varieties for h=(2, 3, 4, ..., n, n)
Hessenberg varieties are defined to be certain subvarieties of the flag variety. By the works of Balibanu–Crooks and Horiguchi–Masuda–Shareshian–Song, it is known that the topology of a regular Hessenberg variety for h=(2, 3, 4, ..., n, n) is closely related to the topology of the toric orbifold associated to a partitioned permutohedron.
In this talk, I will explain that this relation can also be seen in terms of their divisors. This is an ongoing project with Haozhi Zeng.
Classification of toric manifolds of Picard number 4
The explicit classification of complete non-singular toric varieties with small Picard numbers is a fundamental problem in toric geometry. Kleinschmidt classified those with Picard number 2 in 1988, and Batyrev extended the classification to Picard number 3 in 1991. Building on this work, we provide a complete classification of complete non-singular toric varieties with Picard number 4.
Cohomology bases of toric surfaces
The rational cohomology ring of a compact toric surface can be described in two ways: one in terms of intersection products of Weil divisors and the other in terms of cup products of cohomology classes associated with specific cells. In this talk, we investigate the relationship between these two descriptions. Specifically, we introduce two cohomology bases, the Poincaré dual basis and the cellular basis, and demonstrate that the matrices representing the intersection and cup products with respect to these bases are inverses of each other. This is joint work with Tseleung So and Jongbaek Song.
On toric degenerations of flag varieties of finite type
A flag variety is defined as the homogeneous space G/B, which is a smooth projective variety, where G is a semisimple algebraic group over ℂ and B is a Borel subgroup. Although a flag variety is not necessarily toric, it admits toric degenerations. Recently, Fujita and Oya provided a family of toric degenerations arising from cluster structures. When the number of seeds in a given cluster structure is finite, we say that the cluster structure is of finite type. In this talk, we consider a family of toric degenerations of flag varieties of finite type arising from cluster structures. In particular, we analyze toric degenerations of the flag variety Flag(ℂ5) of cluster type D6. This talk is based on joint work with Eunjeong Lee.
Bott manifolds of Bott–Samelson type and assemblies of ordered partitions
Bott manifolds are smooth projective toric varieties providing interesting avenues among topology, geometry, representation theory, and combinatorics. They are used to understand the geometric structure of Bott–Samelson varieties, which provide desingularizations of Schubert varieties. However, not all Bott manifolds originate from Bott–Samelson varieties. Those that do are specifically referred to as Bott manifolds of Bott–Samelson type. In this talk, we explore a relationship between Bott manifolds of Bott–Samelson type and assemblies of ordered partitions. This talk is based on joint work with Junho Jeong and Jang Soo Kim.
Signatures of symmetric matrices defined by plane vector sequences
Motivated by the work by Fu–So–Song: The integral cohomology ring of four-dimensional toric orbifolds (arXiv:2304.03936), one can define a symmetric matrix from a plane vector sequence. In this talk, I will discuss its signature.
Steenrod operations for 4-dimensional toric orbifolds
A toric orbifold X is characterized by a simple polytope P and a characteristic function λ. A fundamental problem is to understand the relationship between the topology of X, the combinatorial data from the pair (P, λ), and the algebraic structure of its cohomology H*(X).
In this talk, I will present my recent work exploring how the pair (P, λ) determines the Steenrod operations on the cohomology of a 4-dimensional toric orbifold X. I will also discuss applications of Steenrod operations to computing the stable splittings and gauge groups of X, and establishing a criterion for the existence of a spin structure when X is smooth.
Combinatorics on partitioned permutohedra
A permutohedron is the convex hull of the orbit of a generic point in a vector space under the natural action of the symmetric group. By intersecting a permutohedron with hyperplanes orthogonal to simple roots, one can construct a new class of polytopes known as partitioned permutohedra. In this talk, we will briefly review topological motivations behind these polytopes and explore their f-vectors, h-vectors, and γ-vectors. This talk is based on joint works with T. Horiguchi, M. Masuda, T. Sato, and J. Shareshian.
Type B real permutohedral varieties
We focus on type B real permutohedral varieties, which are real toric varieties associated with signed permutohedra. These are analogous to real permutohedral varieties, which are associated with permutohedra. The rational cohomology of a real permutohedral variety is known to be fully described in terms of alternating permutations. In this talk, we provide an explicit description of the rational cohomology of a type B real permutohedral variety in terms of B-snakes, the type B analogs of alternating permutations.
Peterson varieties and Blume toric orbifolds
The Peterson variety is introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. In this talk, we discuss an explicit morphism from the Peterson variety to the Blume toric orbifold. As an application, we calculate the totally nonnegative part of the Peterson variety in Lie type A and give a proof of Rietsch’s conjecture. This is joint work with Hiraku Abe.