KPPY 75

Let $H$ be a finite group. We denote by $\mathbb{Q}H$ the group algebra of $H$ over the rational number field $\mathbb{Q}$. For a nonempty subset $T \subseteq H$, we set $\underline{T}:= \sum_{t \in T}t$ which is treated as an element of $\mathbb{Q}H$. A subalgebra $\mathcal{A}$ of the group algebra $\mathbb{Q}H$ is called a Schur ring (briefly S-ring ) over $H$ if there exists a partition $\mathrm{Bsets}(\mathcal{A}):= \{ T_0, T_1, \dotsc, T_r \}$ of $H$ satisfying the following conditions:

1. $\{ \underline{T_i} \mid T_i \in \mathrm{Bsets}(\mathcal{A}) \}$ is a linear basis of $\mathcal{A}$;
2. $T_0 = \{ 1_H\}$;
3. $T_i^{-1}:=\{ t^{-1} \mid t \in T_i \} \in \mathrm{Bsets}(\mathcal{A})$ for all $T_i \in \mathrm{Bsets}(\mathcal{A})$.

In this talk, we consider the Schurity problem of non-commutative $p$-Schur rings over groups of order $p^3$. In particular, we give a family of non-Schurian $p$-Schur rings over groups of order $p^3$.

In this talk, I introduce metrices on finite directed graphs, very recently developed by Etzion and Firer. This concept is a generalization of poset metrics introduced by Brualdi, et.al. The related problem is also presented.

A Cayley map is a 2-cell embedding of a Cayley graph in orientable surface with the same orientation at each vertex. The concept of a skew-morphism generalizes several concepts previously studied with respect to regular Cayley maps, and allows for unified theory of regular Cayley maps and their automorphism groups. [R. Jajcay, J. Siran, Skew-morphisms of regular Cayley maps, Discrete Mathematics 244 (2002), 167-179] In this talk, we give an overview of the Skew-morphisms of regular Cayley maps, and present recent results.

In this presentation, I introduce Euler's conjecture for latin squares. Furthermore, I want to introduce some examples of latin squares and an open problem of latin hypercubes.

In this talk we will introduce a combinatorial interpretation for q-integrals. Then we will show that Askey's q-Selberg integral can be restated as the generating function for the reverse plane partitions contained in a square with certain weights. As a special case we obtain the well known "trace" generating function for the reverse plane partitions.