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

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......