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

Lin Zhang

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......

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

Abstract

The integral formulae pertaining to the unitary group 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......

Paper Structure

This paper contains 28 sections, 60 theorems, 530 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Schur-Weyl duality
  3. Matrix integrals over unitary groups
  4. The case where $k=1$
  5. The case where $k=2$
  6. The general case
  7. The case where $k=3$
  8. The generalized Schur-Weyl duality
  9. Some matrix integrals related to random matrix theory
  10. Discussion and concluding remarks
  11. Appendix
  12. Partitions
  13. Young frames and Young tableaux
  14. Permutations
  15. Products of power sums
  16. ...and 13 more sections

Key Result

Proposition 2.2

Let $|\psi\rangle = \sum_{i,j=1}^{d_A,d_B}\gamma_{ij}|a_i b_j\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ be a vector in the tensor product of two Hilbert spaces, where $\{|a_i\rangle\}_{i=1}^{d_A}$ is an orthonormal basis for $\mathcal{H}_A$ and $\{|b_i\rangle\}_{i=1}^{d_B}$ is an orthonormal f holds, with positive real coefficients $\lambda_k$. The $\lambda_k$ are uniquely determined as the

Theorems & Definitions (129)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6: The dual theorem
  • proof
  • ...and 119 more