The orthogonal Weingarten formula in compact form

Abstract

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity $I(i_1,...,i_{2k}:j_1,...,j_{2k}) = \int_{O_n}u_{i_1j_1} ... u_{i_{2k}j_{2k}} du$ replaced by the more advanced quantity $I(a)=\int_{O_n}\Pi u_{ij}^{a_{ij}} du$, depending on a matrix of exponents $a\in M_n(\mathbb N)$. Among consequences, we establish a number of basic facts regarding the integrals $I(a)$: vanishing conditions, possible poles, asymptotic behavior.