Mini-Workshop on Discrete Mathematics and Related Topics
Mini-Workshop on Discrete Mathematics and Related Topics
Dates: 12th September
Venue: Waseda University, Nishi-Waseda campus, Conference room 2 on the 1st floor of the north tower of Building 55
Contact: Tsuyoshi Miezaki miezaki (at) waseda.jp
Organizers: Tsuyoshi Miezaki (Waseda University), Norihiro Nakashima (Nagoya Institute of Technology), Manabu Oura (Kanazawa University)
Program:
12th September
9:30–10:10: Ryo Uchiumi (The University of Osaka)
The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups
In this talk, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod q hyperplane arrangements. We will show that its character is a quasi-polynomial in q and can be expressed as a sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials. If time permits, we will discuss the results for the Coxeter arranegements with Weyl group actions.
10:20–11:00: Masato Konoike (The University of Osaka)
Characteristic quasi-polynomials of restrictions of Shi arrangements of type D
Characteristic quasi-polynomials enumerate the number of points in the complement of hyperplane arrangements modulo positive integers. In 2024, Higashitani and Nakashima computed the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type B by one given hyperplane. In this talk, we extend their work to type D: we compute the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type D by one given hyperplane. Furthermore, we determine whether period collapse occurs in the characteristic quasi-polynomials of the deletions of the Shi arrangements of type D.
11:20–12:00: Takuro Abe (Rikkyo University)
Double poins of projective line arrangements and algebra
The famous Sylvester-Gallai theorem asserts that every non-pencil real projevtive line arrangements have at least one double points, but this is no more true when the base field is the complex number field. Anzis and Tohaneanu conjectured that several complex version of this for a supersolvable arrangements, which was solved by the speaker. We give an overview of them and introduce recent developments around this double point problem.
14:00–14:40: Sarbaini (Kanazawa University)
Centralizer Algebras of Two Permutation Groups of Order 1344
There are two permutation groups that they share the same character table of order $1344$. We take up natural representations on 8 and 14 letters respectively. The purpose of this paper is to examine the semi-simple structure of centralizing algebras in the tensor representation.
14:50–15:30: A.K.M. Selim Reza (Kanazawa University)
Computation of Vector-Valued Invariants for a Finite Complex Reflection Group
We consider the complex reflection group \( H_1 \), identified as No. 8 in the Shephard-Todd classification. In this paper, we present computations of the vector-valued invariants associated with various representations of \( H_1 \). Additionally, we investigate the structure of the corresponding invariant rings.
15:50–16:30: Tongyu Nian (The University of Osaka)
q-deformation of graphic arrangements
We first observed a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field Fq . These two arrangements are defined by the determinants of the Vandermonde and the Moore matrix, respectively. These two matrices are transformed to each other by replacing a natural number n with q^n (q-deformation). In the talk, I will introduce the notion of “q-deformation of graphical arrangements” and extend it to simplicial complexes, followed by a main theorem concerning the q-analog. This new class of arrangements enjoy the similarities to graphical arrangements. We show that many invariants of the “q-deformation” behave as the ones of the graphical arrangements. This is a joint work with Shuhei Tsujie, Ryo Uchiumi and Masahiko Yoshinaga.
16:40–17:20: Himadri Shekhar Chakraborty（Shahjalal University of Science and Technology）
Higher weight generalization of Jacobi polynomials for codes
In this talk, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between the resulting Jacobi polynomials. Moreover, we present the Jacobi analogue of MacWilliams identity for both higher and extended weight enumerators. We also present that the higher Jacobi polynomials for linear codes whose subcode supports form $t$-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique. Finally, we demonstrate how higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials. This is a joint work with Tsuyoshi Miezaki.