Frame Numbers and Jacobson Radicals for Partial Geometries and Related Coherent Configurations

TL;DR

This paper investigates the modular representation theory of rank 3 association schemes from partial geometries, deriving formulas for Frame numbers, analyzing Jacobson radicals, and exploring the algebraic structure over finite fields.

Contribution

It provides explicit formulas for Frame numbers, characterizes primes affecting semisimplicity, and describes the Jacobson radical in various cases, advancing understanding of modular representations of these schemes.

Findings

Derived a closed formula for the Frame number.

Characterized primes where the adjacency algebra is not semisimple.

Analyzed the Jacobson radical and p-ranks in specific cases.

Abstract

We study the modular representation theory of rank $3$ association schemes arising from partial geometries with parameters $(s,t,\alpha)$. First, we obtain an explicit closed formula for the Frame number of the point scheme in terms of the number of points $v$ and the parameter $s+t+1-\alpha$, and use it to characterize the primes $p$ for which the adjacency algebra over $\mathbb{F}_p$ is not semisimple. We then give a complete case-by-case description of the Jacobson radical of this algebra in four arithmetic situations and determine the generic $p$-ranks of the adjacency matrices. As a step toward understanding the modular representation theory of coherent configurations of type $[3,2;3]$ associated with strongly regular designs, we analyze the relationship between the modular structure of the point scheme and that of the design algebra. For the generalized quadrangle $\mathrm{GQ}(2,2)$ we obtain partial results on the structure of the $2$-modular adjacency algebra $\mathbb{F}_2 \mathfrak{X}$, and we explain the representation-theoretic difficulties that prevent a complete determination of its Wedderburn decomposition and Gabriel quiver, which remains open and is formulated as Problem~6.8.