The Irreducible Modules of the Terwilliger Algebras of Doob Schemes

Let Y be any commutative association scheme and we fix any vertex x of Y . Terwilleger introduced a non-commutative, associative, and semi-simple C-algebra T = T (x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra of Y with respect to x . Let W (⊂ C|X |) be an irreducible T (x)-module. W is said to be thin if W satisfies a certain simple condition. Y is said to be thin with respect to x if each irreducible T (x)-module is thin. Y is said to be thin if Y is thin with respect to each vertex in X . The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphs K4. Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible T (x)-modules of Doob schemes.