Geometric interpretation of the cumulants for random matrices previously defined as convolutions on the symmetric group

We show that, dealing with an appropriate basis, the cumulants for N x N random matrices (A(1),...., A(n)), previously defined in [2] and [3], are the coordinates of E {Pi(A(1) circle times center dot center dot center dot circle times A(n))}, where Pi denotes the orthogonal projection of A(1)circle times center dot center dot center dot circle times A(n) on the space of invariant vectors of M(N)(circle times n) under the natural action of the unitary, respectively orthogonal, group. In this way we make the connection between [5] and [2], [3]. We also give a new proof in that context of the properties satisfied by these matricial cumulants.