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-algebraT=T(x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra ofY with respect to x. Let $$W( \subset C^{|X|} ) $$ 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 graphsK 4 . 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.