Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification

TL;DR

This paper classifies groups whose association schemes produce almost commutative Terwilliger algebras, showing they are either abelian or Camina groups, and computes algebraic properties for these cases.

Contribution

It provides the first classification of groups with almost commutative Terwilliger algebras in association schemes, identifying abelian and Camina groups as key cases.

Findings

All such groups are either abelian or Camina groups.

Computed dimensions and primitive idempotents for these algebras.

Classification covers the first three group types; the final case is deferred.

Abstract

Terwilliger algebras are a subalgebra of a matrix algebra that are constructed from association schemes over finite sets. In 2010, Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative. In that paper she gave five equivalent conditions for a Terwilliger algebra to be almost commutative. In this paper, we provide a classification of which groups result in an almost commutative Terwilliger algebra when looking at the group association scheme (the Schur ring generated by the conjugacy classes of the group). In particular, we show that all such groups are either abelian, or Camina groups. Following this classification, we then compute the dimension and non-primary primitive idempotents for each Terwilliger algebra of this form for the first three types of groups whose group association scheme gives an almost commutative Terwilliger algebra. The final case will be considered in a second paper.