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

Abstract

We show that, dealing with an appropriate basis, the cumulants for N×N random matrices (A ₁,…, A _n), previously defined in [2] and [3], are the coordinates of $\mathbb{E}\left\{ {\prod \left( {{A_1} \otimes \cdots \otimes {A_n}} \right)} \right\},$where II denotes the orthogonal projection of A ₁⊗…⊗A _n on the space of invariant vectors of M ^⊗n _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.