Characterization of Balanced Coherent Configurations

TL;DR

This paper explores the structure of balanced coherent configurations derived from group actions, focusing on their combinatorial properties, primitive idempotents, and classification of reduced schemes.

Contribution

It introduces the concept of balanced schemes, analyzes their primitive idempotents, and characterizes reduced balanced schemes using combinatorial methods.

Findings

Balanced schemes have specific central primitive idempotents.

The paper characterizes reduced balanced schemes.

It provides a combinatorial approach to classifying schemes.

Abstract

Let $G$ be a group acting on a finite set $\Omega$. Then $G$ acts on $\Omega\times \Omega$ by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of $G$ on $\Omega_i\times \Omega_j$ is constant whenever $\Omega_i$ and $\Omega_j$ are orbits of $G$ on $\Omega$. One can conclude from the assumption that the actions of $G$ on ${\Omega_i}$'s have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes.