講演タイトルと概要

Anton Ayzenberg (Higher School of Economics)

Towards periodic permutohedral variety

概要：The construction of twins gives a relation between smooth manifolds of isospectral Hermitian matrices having staircase form and regular semisimple Hessenberg varieties. Now we take a generic spectrum and consider the manifold of periodic tridiagonal isospectral matrices of size n. If we consider the twin of this manifold, we get an interesting smooth submanifold Y in a flag variety. The manifold Y contains n copies of a permutohedral variety as some sort of divisors, and, probably, it may be related to the infinite flag manifolds, Kac-Moody algebras and other exciting things. Although I don't have any particular results about the manifold Y, I believe it is worth studying and hope for discussion.

榎園 誠 (東京理科大学)

Uniform bases for ideal arrangements

概要：Ideal arrangements are subarrangements of the Weyl arrangement defined by lower ideals in the root system. The logarithmic derivation modules of such arrangements are related to the cohomology rings of regular nilpotent Hessenberg varieties by Abe-Horiguchi-Masuda-Murai-Sato's work. In this talk, we introduce the notion of uniform bases for ideal arrangements and discuss the existence and the construction of uniform bases in all Lie types. As an application, we explain an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types. This is joint work with Tatsuya Horiguchi, Takahiro Nagaoka and Akiyoshi Tsuchiya.

藤田 直樹 (東京大学)

Classification of weak Fano Hessenberg varieties

概要：A regular semisimple Hessenberg variety in type A is a nonsingular closed subvariety of a flag variety, which generalizes both the flag variety and the permutohedral variety. Unlike the flag variety, the permutohedral variety is not Fano in general, but weak Fano. Hence weak Fano Hessenberg varieties also generalize these varieties. Using the theory of degeneracy loci, Anderson-Tymoczko related the cohomology classes of regular semisimple Hessenberg varieties with Richardson varieties. We use this relation to give a necessary and sufficient condition for a regular semisimple Hessenberg variety to be weak Fano. This is joint work with Hiraku Abe and Haozhi Zeng.

原田芽ぐみ (McMaster University)

The cohomology of Hessenberg varieties and the Stanley-Stembridge conjecture

概要：Hessenberg varieties are subvarieties of the flag variety with rich connections to algebraic geometry, combinatorics, and representation theory. They are important examples in combinatorial algebraic geometry, in the sense that much of their geometry and topology can be described in Lie-theoretic and combinatorial terms. Special cases of Hessenberg varieties include famous classes of varieties studied in other contexts, such as Springer fibers, Peterson varieties, and the permutohedral variety.

This lecture series will focus on one area within the study of Hessenberg varieties, namely, the symmetric group representation on the cohomology rings of regular semisimple Hessenberg varieties, as defined by Tymoczko, and the ensuing connection between Hessenberg varieties and the long-standing Stanley-Stembridge conjecture in combinatorics. Some related questions regarding the construction of explicit permutation bases will also be briefly discussed. The lectures will consist of three parts:

• (a) An introduction to Hessenberg varieties.

• (b) The Stanley-Stembridge conjecture.

• (c) Recent developments.

この集中講義は大阪市立大学の幾何構造論特別講義Ⅰ・Ⅱとして行われます．

Jaehyun Hong (Korea Institute for Advanced Study)

Positivity of chromatic symmetric functions associated to Hessenberg functions of bounce number 3

概要：In this talk we consider chromatic symmetric functions of the natural unit interval graphs associated with Hessenberg functions. We show that if the maximal length of chain is 3, or equivalently, the Hessenberg function has bounce number 3, then the corresponding chromatic symmetric function is expanded as a positive linear combination of elementary symmetric functions, using the description of its expansion by Schur functions given by Gasharov. This is joint work with S. Cho.

Eunjeong Lee (Institute for Basic Science)

Torus orbit closures in the flag variety

概要：Let $G$ be a simple algebraic group of classical Lie type over $\mathbb{C}$, and $B$ a Borel subgroup. The full flag variety $G/B$ has an action of a maximal torus $T$ induced by the left multiplication. The closure of a $T$-orbit in $G/B$ is a toric variety. It has been known that for a \textit{generic} point $p \in G/B$, the toric variety $\overline{T \cdot p}$ is defined by the Weyl chambers of $G$. When $G = \mathrm{SL}_{n}(\mathbb{C})$, then the torus orbit closure of a generic point is the permutohedral variety, which is a regular semisimple Hessenberg variety. In this talk, we study torus orbit closures $\overline{T \cdot p}$ for not necessarily generic $p$. More precisely, we first observe torus orbit closures in $G/B$ for $G = \mathrm{SL}_{n}(\mathbb{C})$. And then, we study torus orbit closures in $G/B$ for $G$ of classical Lie type. For any $p \in G/B$, we introduce three kinds of \textit{retractions} on the Weyl group $W$ of $G$ to the set of fixed points $(\overline{T \cdot p})^T$ which can be used to describe the fan of the toric variety $\overline{T \cdot p}$. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.

John Shareshian (Washington University in St. Louis)

Two conjectures around regular semisimple Hessenberg varieties.

Abstract: I will discuss the two following conjectures:

1) Ayzenberg and Buchstaber studied manifolds of isospectral Hermitian matrices. They showed that these manifolds admit a torus action to which the theory of Goresky, Kottwitz and MacPherson can be applied. Each of the associated moment graphs admits action of an appropriate symmetric group, from which arises a representation on cohomology. In work with Andy Wilson, we conjecture that the Frobenius characteristic of this representation is a unicellular LLT polynomial.

2) Regular semisimple Hessenberg varieties admit torus actions to which the GKM theory mentioned in (1) also applies. This leads to Weyl group representations on cohomology. In joint work with Patrick Bronsan, we conjecture that knowing the characters of these representations is the same as being able to count rational points on naturally corresponding regular semisimple Hessenberg varieties defined over finite fields.

辻栄 周平 (広島国際学院大学)

The $e$-positivity and forbidden induced subgraphs

概要：In the first part of this talk, we will focus on the $e$-positivity of graphs in terms of forbidden induced subgraphs and introduce recent progress. In the second part, using Dahlberg's sign-reversing involution method, we will produce some new $e$-positive graphs.