For a matrix *-algebra B, consider the matrix *-algebra A consisting of the
symmetric tensors in the n-fold tensor product of B. Examples of such algebras
in coding theory include the Bose-Mesner algebra and Terwilliger algebra of the
(non)binary Hamming cube, and algebras arising in SDP-hierarchies for coding
bounds using moment matrices. We give a computationally efficient block
diagonalization of A in terms of a given block diagonalization of B, and work
out some examples, including the Terwilliger algebra of the binary- and
nonbinary Hamming cube. As a tool we use some basic facts about representations
of the symmetric group.Comment: 16 page