Workshop on Combinatorial topics in Shinshu 2022

The purpose of this workshop is to exchange information and exchange with young researchers and students who are interested in combinatorics and related topics.

Because of preventative measures against coronavirus, the number of participants is limited. If you want to attend to this workshop, please contact Yazawa, Akiko (yazawa [at] math.shinshu-u.ac.jp) in advance. We appreciate your understanding and cooperation.

Dates

2022/07/25--27. (for three days)

Venues

We use two venues for the workshop.

Speakers

Program

2022/7/25 Mon.

Meeting room (M2, Matsumoto Performing Arts Centre) will be open before 9:45.
10:00 -- 11:30
Akiko Yazawa (Shinshu University)
The Hessian matrices of generating polynomials associated to graphs and matroids
We consider the generating polynomials for matroids and graded Artinian Gorenstein algebras defined by the polynomials. We study the strong Lefschetz property of the algebras and strictly log-concavity of the polynomials. We show their properties by using the Hessian matrices of the polynomials. This work is a joint work with Satoshi Murai and Takahiro Nagaoka.
13:30 -- 15:00
Zixuan Wang (Hokkaido University)
On Free Deformations of Graphic Arrangements and Free Coxeter Multiarrangements
Graphic arrangements can be thought of as the subarrangements of Coxeter arrangements of type A. It is well known that Coxeter arrangements are free. Some researchers classify the free graphic arrangements, and give the graphical interpretation of the exponents of free graphic arrangements. We are interested in the free deformations of graphic arrangements. We consider the subarrangements $\operatorname{Shi}(G)$ of Shi arrangements associated with a graph $G$. Studying the freeness of multiarrangements is a great way to study freeness of hyperplane arrangements. Many researchers try to classify the free Coxeter multiarrangements. It is a hard work. Even the free multiplicities of Coxeter arrangements of type A have not been solved. I will introduce some free Coxeter multiarrangements and their multiderivations.
15:30 -- 17:00
Takuya Saito (Hokkaido University)
Discriminantal arrangements, Derived matroids, and their combinatorics
In 1989, Manin and Schectman introduced discriminantal arrangements, a hyperplane arrangement with a higher pure braid group as the fundamental group of the complement. On the other hand, definitions equivalent to discriminantal arrangements, e.g. derived matroids, have been given in various contexts. In this presentation, we will talk about the relationship between derived matroids and discriminantal arrangements. After that, we classify Discriminantal arrangements determined from arrangements with 6 hyperplanes in general positions in dimensions 2 and 3. If time enables, the relationship with adjoints of a matroid will also be presented.

2022/7/26 Tue.

Meeting room (meeting room 2, Nakamachi Kurassic-kan) will be open before 9:20.
9:30 -- 11:00
Kosuke Shibata (National Institute of Technology, Yonago College)
Explicit construction of a minimal free resolution of the Specht ideal of shape $(n−d,d)$
Specht modules are crucial in the representation theory of symmetric groups, and are spanned by Specht polynomials. They are indexed by partitions of integers, and in characteristic 0 the Specht modules form a complete list of irreducible representations of the symmetric group. On the other hand, a Specht ideal is the ideal of a polynomial ring generated by Specht polynomials. These ideals have been studied from various points of view (e.g. combinatorial commutative algebra, subspace arrangements). In this talk, we give an explicit construction of minimal free resolutions of Specht ideals of shape $(n-d,d)$ when the characteristic is 0.
14:00 -- 15:30
Hiroki Mizuno (Shinshu University)
Introduction to the Kawazumi-Kuno fomula of Dehn twists
In low-dimensional topology, Dehn twist is a type of self-homeomorphism on a surface and plays an important role in a description about symmetries of surfaces. For example, the mapping class group, a discreat group corresponding to symmetries of surfaces, is generated by Dehn twists. In this presentation, we introduce an explicit formula, shown by Kawasumi-Kuno,of Dehn twist on the completed group ring.
16:00 -- 17:30
Yu Tajima (Hokkaido University)
Magnitude homology of graphs and the homotopy type of the Asao-Izumihara complexes
The magnitude is an invariant for metric spaces defined by Leinster. The magnitude homology is defined by Hepworth and Willerton as a categorification of the magnitude. Recently, Asao and Izumihara introduced CW complexes whose homology groups are isomorphic to direct summands of graph magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs defined by Y. Gu (and some other graphs). We use the discrete Morse theory for proofs. This is a joint work with Masahiko Yoshinaga (Osaka University).

2022/7/27 Wed.

Meeting room (M2, Matsumoto Performing Arts Centre) will be open before 9:45.
10:00 -- 11:30
Ryosuke Sato (Chuo University)
Characters of (quantum) groups and Markov processes
In this talk, we will discuss Markov processes with representation-theoretic origin. For a given compact (quantum) group, the Fourier analysis produces a correspondence between characters and probability measures on the irreducible representations. Hence, the study of Markov processes on such spaces reduces to the study of time evolutions of characters, and we can obtain some explicit formulas using representation theory.
13:00 -- 14:30
Shogo Sugimoto (Waseda University)
Special Functions in Schubert Calsulus
It is well known that Schur polynomials represent Schubert classes in the cohomology ring of Grassmann variety. Pragacz has proved that Schur $P$ functions introduced by Schur represent Schubert classes in the cohomology ring of maximal orthogonal Grassmann variety. Grothendieck polynomials introduced by Lascoux and Schützenberger (rest. $GP$ functions introduced by Ikeda and Naruse) are $K$-theoretic analogues of Schur polynomials (Schur $P$ functions). In this workshop we introduce combinatorial aspects of these functions.
15:00 -- 16:30
JuAe Song (Tokyo Metropolitan University)
Chip firings and rational function semifields of tropical curves
Recently, the speaker found that rational functions called chip firings have essential information about tropical curves and their rational function semifields. Originally, chip firings were a concept on undirected graphs. In this talk, we focus on chip firings and introduce how they played important roles in the tropical curve theory with some significant results.

Acknowledgements

This workshop is held supported by the following:

Organizer

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