The orthogonal Weingarten formula in compact form

Abstract

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity $I(i_1,...,i_{2k}:j_1,...,j_{2k})={\int }_{O_n}{u}_{i_1{j}_1}...u_{i_{2k}{j}_{2k}}du$ 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.

12 pages

References in corpus (5)

• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• On some properties of orthogonal Weingarten functions
• Jucys-Murphy elements and Weingarten matrices
• On polynomial integrals over the orthogonal group
• Jucys-Murphy Elements and Unitary Matrix Integrals

Cited by in corpus (4)

• Jucys-Murphy elements and Weingarten matrices
• On polynomial integrals over the orthogonal group
• On the minimum output entropy of random orthogonal quantum channels
• Commutators of random matrices from the unitary and orthogonal groups