Integrals of Irreducible Representations of Classical Groups

TL;DR

This paper analyzes integrals of matrix element monomials of irreducible classical group representations, exploring dualities and asymptotics relevant to physics fields like quantum mechanics and gauge theory.

Contribution

It introduces new duality theorems and asymptotic formulas for group integrals involving irreducible representations of classical groups.

Findings

Derived duality theorems for classical group integrals

Computed asymptotics as group rank increases

Linked integrals to applications in physics

Abstract

This paper is concerned with integrals which integrands are the monomials of matrix elements of irreducible representations of classical groups. Based on analysis on Young tableaux, we discuss some related duality theorems and compute the asymptotics of the group integrals when the signatures of the irreducible representations are fixed, as the rank of the classical groups go to infinity. These group integrals have physical origins in quantum mechanics, quantum information theory, and lattice Gauge theory.