The Ring of Support-Classes of $\mathrm{SL}\_2(\mathbb F\_q)$

TL;DR

This paper introduces a new subring of the group algebra of SL_2(F_q), constructed by summing elements based on their support, which can be utilized to build association schemes.

Contribution

It defines and analyzes the support-class subring of the group algebra of SL_2(F_q), providing a novel algebraic structure for combinatorial applications.

Findings

Defined the support-class subring of the group algebra.

Showed how this subring can be used to construct association schemes.

Provided algebraic properties and potential applications of the subring.

Abstract

We introduce and study a subring $\mathcal{SC}$ of $\mathbb Z[\mathrm{SL}\_2(\mathbb F\_q)]$ obtained by summing elements of $\mathrm{SL}\_2(\mathbb F\_q)$ according to their support. The ring $\mathcal SC$ can be used for the construction of several association schemes.