Terwilliger algebras of wreath products of association schemes

Abstract

<p>In this thesis, we study the T -algebras of symmetric association schemes that are obtained as the wreath product of H(1, m) for m ≥ 2. We find that the D-class association scheme Kn1&m22;Kn2&m22;&cdots; &m22;KnD formed by taking the wreath product of one-class association schemes Kni = H(1, ni) has the triple-regularity property. We determine the dimension of the T -algebra for the association scheme Kn1&m22;Kn2&m22;&cdots; &m22;KnD . We also show that the wreath power Km&m22;D =Km&m22;Km&m22;&cdots;&m22;Km , D copies of Km, is formally self-dual. We give a complete description of the irreducible T -modules and the structure of T -algebra for Km&m22;D for m ≥ 2 by essentially studying the irreducible modules of 2 copies of Km and then extending it to the general case for D copies of Km. Through these calculations we obtain that the T -algebra for Km&m22;D is MD+1C ⊕C⊕12 DD+1 for m ≥ 3, and MD+1C ⊕C⊕12 DD-1 m = 2.</p>