Triple transitivity of the dicyclic group and structure of its Terwilliger algebra

Abstract

The Terwilliger algebra is a useful structure in the study of association schemes. Bannai and Munemasa showed that the Terwilliger algebra T(G) of the group association scheme of an arbitrary finite group G satisfies the following bounds: T0 ⊆ T(G) ⊆ T(G), where T0 is a certain vector space and T(G) is the centralizer algebra of the permutation representation of G arising from the action of conjugation. The structures of Terwilliger algebras of group association schemes have been determined only in the case of a few families of finite groups. In this paper, we consider the dicyclic group Dic(n)=〈a, b | a2n = e, b2 = an, bab−1 = a−1〉 of order 4n, n ≥ 2, and prove that Dic(n) satisfies a property called triple transitivity that implies T0 = T (Dic(n)) = T(Dic(n)). We then completely determine the structure of T (Dic(n)) by obtaining the Wedderburn decomposition of T(Dic(n)).