Representation theory and central limit theorems for traces of commutators for compact Lie groups

TL;DR

This paper develops a representation-theoretic, combinatorial approach to central limit theorems for traces of products of commutators in compact Lie groups. By leveraging Sundaram–Su up-down tableaux for symplectic and orthogonal groups and Stembridge’s staircase representations for unitary groups, the authors express moments of traces as finite sums over irreducible characters and control them via dimension bounds. The method produces CLTs showing that, as the rank grows, the traces converge to standard normal variables (complex in the unitary case and real in the orthogonal and symplectic cases), with the odd moments vanishing and even moments matching the double factorial pattern. The work foregrounds a coherent, purely representation-theoretic toolkit for these limit theorems, complementing earlier Weingarten-calculus and free-probability approaches, and notes the results’ alignment with known cases while emphasizing the novelty of the approach. The paper also documents two key lemmas with proofs in the preliminaries and discusses the applicability and potential limitations to general word expressions.

Abstract

There has been some work in the literature on limit theorems for the trace of commutators for compact Lie groups. We revisit this from the perspective of combinatorial representation theory.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 4.1