On the Diaconis-Shahshahani Method in Random Matrix Theory

If Γ is a random variable with values in a compact matrix group K, then the traces Tr(Γ^j) (j ∊ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis and Shahshahani computed the joint moments of any fixed number of these traces if Γ is distributed according to Haar measure and if K is one of U_n, O_n or Sp_n, where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer algebras. The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically, the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to a counting problem, which can be solved by elementary means.