Group character averages via a single Laguerre

Abstract Average of exponential TrReX, i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices ${\rm Tr} \left(\prod _i A_{k_i}\right)$ and are thus labeled by weak compositions. The entries of matrices Ak are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial $L_{N-1}^1(z_{k_i})$, what is a considerable simplification.