Coherent configurations and Frobenius structures

TL;DR

This paper establishes a novel algebraic framework linking coherent configurations to Frobenius structures, enabling spectral analysis and generalization of classical theorems from group theory to association schemes.

Contribution

It introduces a representation of coherent configurations as modules over Frobenius structures and extends the notion of admissible morphisms, connecting these configurations to groupoids and $H^*$-algebras.

Findings

Representation of coherent configurations as modules over Frobenius structures

Introduction of dagger Frobenius structures and their properties

Recovery of original configurations via eigenvectors of matrix O

Abstract

We prove that coherent configurations can be represented as modules over Frobenius structures in the category of real nonnegative matrices. We generalize the notion of admissible morphism from association schemes to coherent configurations. We show that the Frobenius structure associated to a coherent configuration can be modified to become a dagger Frobenius structure, and use this to connect the coherent configurations to groupoids and $H^*$-algebras. We examine the properties of the dagger Frobenius structure with respect to admissible morphisms. We introduce the matrix $O$ obtained as the composition of comultiplication and multiplication of the dagger Frobenius structure and prove that we may obtain the valencies of colors, and thus recover the original coherent configuration, as an eigenvector of $O$. In the last part of the paper, we examine the spectrum of $O$ and apply it to generalize the Lagrange theorem from groups to association schemes.