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

• URL: http://arxiv.org/abs/1408.3782v6
• Date: Wed, 30 Oct 2024 07:54:31 GMT
• Title: Matrix integrals over unitary groups: An application of Schur-Weyl duality
• Authors: Lin Zhang,
• Abstract summary: The integral formulae pertaining to the unitary group $mathsfU(d)$ have been comprehensively reviewed. Schur-Weyl duality serves as a bridge, establishing a profound connection between the representation theory of finite groups and that of classical Lie groups.
• Score: 4.927579219242575
• License:
• 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......

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