Weingarten Calculus
Georg Köstenberger
TL;DR
This work addresses the problem of evaluating Haar integrals of polynomials in the entries of unitary matrices. It develops a unified framework that ties together classical results with the Collins–Sniady Weingarten calculus via Schur–Weyl duality and centralizer projections, enabling explicit moment formulas. A key contribution is the explicit Weingarten formula and its expression through Jucys–Murphy elements and Moore–Penrose inverses, culminating in a closed form for the 2n-th moment of Tr(U) as a sum over partitions with length constraint. The results provide a versatile toolkit for higher-mower calculations of unitary-invariant random matrices, with implications for random matrix theory and quantum information theory.
Abstract
We consider the problem of computing the integral $$ \int_{\mathcal{U}(d)} u_{i_1j_1}\cdots u_{i_nj_n} \bar{u}_{i'_1j'_1} \cdots \bar{u}_{i'_{n'}j'_{n'}} dU, $$ where the integration takes place with respect to the probability Haar measure on the unitary group $\mathcal{U}(d)$, and the $u_{ij}$ denotes the $ij$-th entry of a unitary matrix $U$. We present a unified approach connecting classical results, the explicit formula for the integral given by B. Collins and P. Sniady and subsequent works of various authors providing different points of view. Finally we are able to provide an explicit formula for the $2n$-th moment of the trace of a unitary Haar random matrix, generalizing a result of P. Diaconis.