Matrix integrals over unitary groups: An application of Schur-Weyl duality

TL;DR

This paper reviews matrix integrals over unitary groups using Schur-Weyl duality, connecting representation theory and permutation group characters, and discusses the challenges in computing irreducible characters.

Contribution

It provides new proofs and insights into integral formulae over unitary groups via Schur-Weyl duality and highlights the complexities in evaluating permutation group characters.

Findings

Explicit evaluation of Weingarten functions depends on representation dimensions and characters.

Hook-length and hook-content formulas are used for representation dimensions.

A comprehensive understanding of permutation group characters remains challenging due to lack of closed-form formulas.

Abstract

The integral formulae pertaining to the unitary group $\mathsf{U}(d)$ have been comprehensively reviewed, yielding fresh results and innovative proofs. Central to the derivation of these formulae lies the employment of Schur-Weyl duality, a classical and powerful theorem from the representation theory of groups. This duality serves as a bridge, establishing a profound connection between the representation theory of finite groups (or permutation groups) and that of classical Lie groups, specifically the unitary groups. From the perspective of Schur-Weyl duality, it becomes evident that the computation of matrix integrals over the unitary group is intricately intertwined with the so-called Weingarten function. The explicit evaluation of this function is heavily dependent on three crucial aspects: firstly, the dimensions of the irreducible representations of the unitary groups; secondly, the dimensions of the irreducible representations of permutation groups; and thirdly, the irreducible characters of permutation groups. For the first two aspects, we can rely on well-established formulae. Specifically, the dimensions of irreducible representations of both unitary and permutation groups can be determined using the hook-length formula attributed to Frame, Robinson,and Thrall, as well as the hook-content formula proposed by Stanley. However, the third aspect poses a more intricate challenge. Unfortunately, despite significant efforts, there remains no unifying closed-form formula for the generic irreducible characters of permutation groups, except for a few special cases involving particular partitions. Given the significance of these irreducible characters, it is crucial to have a comprehensive understanding of them. Fortunately, all the information pertaining to the irreducible characters belonging to a given permutation group is encoded in a so-called character table......