Hessenberg 集会 2018 in Osaka
講演タイトルと概要
Hiraku Abe (Osaka Prefecture University)
Introduction to Hessenberg varieties
概要：I will give a brief survey on Hessenberg varieties. The goal of this talk is to stimulate the discussion among participants. Questions/comments are always welcome.
Takuro Abe (Kyushu University)
Hessenberg varieties and hyperplane arrangements
概要：The most intensively studied Hessenberg varieties are regular nilpotent and regular semisimple case. We show that their cohomology groups have presentations in terms of the corresponding hyperplane arrangements and its logarithmic derivation modules. This generalizes Borel's isomorphism between the coinvariant algebra and the cohomology group of flag varieties. This is a joint work with T. Horiguchi, M. Masuda, S. Murai and T. Sato.
Anton Ayzenberg (National Research University Higher School of Economics)
Torus actions on manifolds of isospectral Hermitian matrices
概要：We consider the space Mn of all Hermitian n × n-matrices with the given simple spectrum. Conjugation by diagonal unitary matrices defines the torus action on this space, and this space is equivariantly diffeomorphic to the variety of complete complex flags. We consider the subspaces in Mn which of matrices having zeroes at prescribed positions. All these subspaces are preserved by the torus action. This construction provides a wide variety of interesting examples, related to many areas of mathematics: from dynamical systems to combinatorial geometry.
Vladislav Cherepanov (National Research University Higher School of Economics)
The torus actions of complexity 1 and manifolds of isospectral Hermitian matrices.
概要：In a recent paper, A. Ayzenberg proved that for an effective action of a compact (n-1)-torus on a smooth 2n-manifold, the quotient space is a topological manifold given that the action is in general position. Subsequently, Y. Karshon and S. Tolman showed that for Hamiltonian actions of tori in general position the quotient is homeomorphic to a sphere. We will discuss the generalizations of the results to the case of actions not in general position. As the examples of such actions, manifolds of isospectral Hermitian matrices and their connection to Hessenberg varieties will be discussed.
Naoki Fujita (Tokyo Institute of Technology)
Algebro-geometric aspects of regular Hessenberg varieties and their families
概要：In this talk, we study algebro-geometric aspects of regular Hessenberg varieties in general Lie type. We consider the flat family of regular Hessenberg varieties, and show that the scheme-theoretic fibers over the closed points are all reduced. We also discuss what data of a regular Hessenberg variety are constant along this family, i.e., independent of the choice of a regular element. Such data are expected to be described in terms of the root system. This is joint work with Hiraku Abe and Haozhi Zeng.
Tatsuya Horiguchi (Osaka University)
The problem of a basis of the cohomology ring of a regular nilpotent Hessenberg variety
概要：In this talk, we want to explain a basis of the cohomology ring of a regular nilpotent Hessenberg variety in terms of the root system in type A. This is joint work with Makoto Enokizono, Takahiro Nagaoka, Akiyoshi Tsuchiya in progress.
Takeshi Ikeda (Okayama Universityof Science)
K-theoretic Peterson isomorphism and its applications
概要：I will give a dictionary connecting quantum and affine Schubert calculi in K-theory. As shown in my previous work with Iwao and Maeno, it is just a reinterpretation of the unipotent solution of the relativistic Toda lattice. I am going to discuss some combinatorial outcome of the correspondence.
Anatol Kirillov (RIMS)
On generalized cohomology theories of nilpotent Hessenberg varieties.
概要：Basically, as the main technical example, I will talk about noncommutative approach to the study of various cohomology theories of complete flag varieties, including LR numbers, Chern classes,... . This approach is based and generalizes a description of the cohomology ring of complete flag varieties in terms of certain difference operators given by C. Dunkl in 1983. As a next step, I will try to explain how to apply this approach to the case of nilpotent Hessenberg type varieties. This step is rather technical identification of our results with results concerning (regular nilpotent) Hessenberg varieties which were obtained during the last 30 years in Japan, USA, France,... .
Tomoo Matsumura (Okayama University of Science)
Degeneracy loci and Hessenberg varieties in type C
概要：In type A, it is known that the Hessenberg varieties can be realized as the degeneracy loci associated to dominant permutations so that their cohomology classes are given by certain specialization of the corresponding double Schubert polynomials under some assumptions. It is natural to ask if similar facts can be obtained for other classical types. In this talk, I will review basic facts for the type C degeneracy loci and discuss the relation of their cohomology classes to Hessenberg classes.
Satoshi Murai (Waseda University)
Is there a good additive basis for cohomology rings of regular nilpotent Hessenberg variety
概要：In this talk, I want to discuss possible candidates of additive basis for cohomology rings of regular nilpotent Hessenberg variety. This talk mainly consists of questions, and will not contain any new result.
Masashi Noji and Kazuaki Ogiwara (Osaka City University)
Smooth torus orbit closures in the Grassmannians
概要：It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We prove that simple matroid polytopes are products of simplices and smooth torus orbit closures in the Grassmannians are products of complex projective spaces. Moreover, it turns out that the smooth torus orbit closures are uniquely determined by the corresponding simple matroid polytopes. In this talk, we will explain an idea of the proof.
SeonJeong Park (Ajou University)
Toric Richardson varieties
概要：Given v,w ∈ Sn with v ≤ w, the Richardson variety Xwv is the intersection of the Schubert variety Xw and the opposite Schubert variety Xv. A Bruhat interval polytope Qv,w is the convex hull of all permutation vectors x = (x(1),x(2),...,x(n)) with v ≤ x ≤ w. A Richardson variety is not toric in general and Tsukerman and Williams showed that Xwv is toric if and only if Qv,w is of dimension l(w) − l(v). In this talk, we discuss torus orbit closures in a Richardson variety. We are particularly interested in toric Richardson varieties. This is joint work with Eunjeong Lee and Mikiya Masuda in progress.
Takashi Sato (Osaka City University)
On coinvariant rings of pseudo-reflection groups
概要：The cohomology ring of a regular nilpotent Hessenberg variety is the quotient ring of the corresponding polynomial ring by some ideal. Abe, Horiguchi, Masuda, Murai, and I showed that this ideal is described in terms of hyperplane arrangements. This result is a generalization of the relationship between the coinvariant ring of the Weyl group and the Weyl arrangement. Another way of generalization is that from Weyl groups to pseudo-reflection groups. In this talk, I will give a description of coinvariant rings of pseudo-reflection groups in terms of hyperplane arrangements.
Haozhi Zeng (Huazhong University of Science and Technology)
On Fano and weak Fano Hessenberg varieties
概要：Regular Hessenberg varieties are a family of subvarieties of the full flag variety G/B. This family contains the full flag variety, Peterson variety and perutohedral variety. In this talk, we discuss the Fano and weak Fano regular semisimple Hessenberg varieties in type A. We describe a Fano regular semisimple Hessenberg variety in terms of its Hessenberg function and give a partial description for weak Fano regular semisimple Hessenberg varieties. This is joint work with Hiraku Abe and Naoki Fujita.