Some matrices associated with the split decomposition for a Q-polynomial distance-regular graph

We consider a Q-polynomial distance-regular graph Gamma with vertex set X and diameter D >= 3. For mu, nu is an element of {down arrow, up arrow} we define a direct sum decomposition of the standard module V = CX, called the (mu, nu)-split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix b, beta is an element of C such that b not equal 1 and assume Gamma has classical parameters (D, b, alpha, beta) with alpha = b - 1. Under this assumption Ito and Terwilliger displayed an action of the q-tetrahedron algebra boxed times(q) on the standard module of Gamma. To describe this action they defined eight matrices in Mat(X) (C), called A, A*, B, B*, K, K*, Phi, Psi. For each matrix in the above list we compute the transpose and complex conjugate. Using this information we compute the transpose and complex conjugate for each generator of boxed times(q) on V. (C) 2008 Elsevier Ltd. All rights reserved.