Combinatorics on Flag Varieties and Related Topics 2021

Title & Abstract

Lara Bossinger

Title: Gröbner degenerations of Grassmannains and cluster algebras

Abstract: My lectures will be split in two parts: in the first I will explain the cluster structure of Grassmannains and how it allows to construct (toric) degenerations. This part is based on work of Scott from 2006 (who showed that the homogeneous coordinate ring of the Grassmannian with respect to its Plücker embedding is a cluster algebra) and on work of Gross, Hacking, Keel and Kontsevich from 2018 (who showed, among other things, how to construct toric degenerations of cluster varieties). In the second part of the lectures I will explain how to realize the above mentioned toric degenerations as Gröbner degenerations. Moreover, we will see how one particular maximal cone in the Gröbner fan of a Grassmannain (of finite cluster type) captures all cluster toric degenerations. The second part is based on work of myself from 2020 and a joint preprint with Fatemeh Mohammadi and Alfredo Nájera Chávez from 2020.

Yunhyung Cho

Title: A brief introduction to toric varieties and Newton-Okounkov bodies

Abstract: This talk will be a preliminary talk of this workshop and covers the following topics:

- Introduction to toric varieties (definitions, structure theorems, etc..)

- Sheaves of divisors on toric varieties

- Introduction to Newton-Okounkov bodies: in view of the generalization of moment polytopes.

Naoki Fujita

Lecture 1: Introduction to crystal bases I

Abstract: Kashiwara's crystal basis is a combinatorial skeleton of a representation of a semisimple Lie algebra. Through the Borel-Weil theory, it relates geometry of flag varieties with combinatorics of tableaux and polytopes. In the 1st talk, we survey the theory of crystal bases in the case of special linear groups.

Lecture 2: Introduction to crystal bases II

Abstract: In the theory of crystal bases, it is important to give their concrete realizations. Until now, many geometric or combinatorial realizations have been discovered. In the 2nd talk, we discuss some combinatorial realizations using tableaux and polytopes.

Lecture 3: Newton-Okounkov bodies of flag and Schubert varieties I

Abstract: A Newton-Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. The notion of Newton-Okounkov bodies was originally introduced to study multiplicity functions for representations of a semisimple group, and afterward developed as a generalization of toric theory. In the 3rd talk, we survey the theory of Newton-Okounkov bodies and its geometric applications. In the case of flag and Schubert varieties, we also discuss relations between Newton-Okounkov bodies and crystal bases.

Lecture 4: Newton-Okounkov bodies of flag and Schubert varieties II

Abstract: For a specific Newton-Okounkov body of a flag variety, the associated toric degeneration of the flag variety induces semi-toric degenerations of (opposite) Schubert varieties. In the 4th talk, we discuss which Newton-Okounkov bodies have this property. We also see some explicit descriptions of induced semi-toric degenerations.

Megumi Harada

Title: An introduction to Hessenberg varieties and their Newton-Okounkov bodies

Abstract: This expository talk will consist of two parts. The first part will be an introduction to Hessenberg varieties and their relation to many other topics such as Schubert calculus, representation theory, the theory of (quasi)symmetric functions, and combinatorics. There is much interesting work in this area, so this first part will be a "survey" talk, and I will try to give an overall sense of the history and some of the big themes, instead of spending a lot of time on details. In the second portion of the talk, I will recount what is known about Newton-Okounkov bodies of Hessenberg varieties. In fact, not very much is known, so this second part will be shorter, and more speculative. We will close with some open questions which I hope that members of the audience would be interested to answer!

Akihiro Higashitani

Title: Introduction to combinatorial mutations of polytopes and its applications

Abstract: Combinatorial mutations of polytopes were introduced by Akhtar-Coates-Galkin-Kasprzyk in the context of mirror symmetry for Fano varieties. Afterwards, the speaker extended the framework of combinatorial mutations to more general objects. In the first part of the lectures, the details of combinatorial mutations are explained. In the second part, we discuss the combinatorial mutations of poset polytopes and marked poset polytopes.

These can be applied to show the combinatorial mutation equivalences of specific Newton-Okounkov bodies of flag varieties.