Almost Commutative Terwilliger Algebras I: The Group Association Scheme

Abstract: Terwilliger algebras are a subalgebra of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions. In this paper we first determine an equivalent sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative. We then provide a classification of which finite groups result in an almost commutative Terwilliger algebra when looking at the group association scheme determined by the conjugacy classes. In particular, we show that all such groups are either abelian, or Camina groups. We then compute the dimension of each Terwilliger algebra, and we also express each of the group association schemes with an almost commutative Terwilliger algebra as a wreath product of the group schemes of finite abelian groups and $1-$class association schemes. Furthermore, we give the non-primary primitive idempotents for each Terwilliger algebra for those groups.