[Paper Review] Symmetric groups and random matrices

Piotr Śniady|ArXiv.org|Jan 26, 2003

Random Matrices and Applications14 references3 citations

TL;DR

This paper establishes an exact combinatorial correspondence between products of conjugacy class indicators in symmetric groups and moments of Jucys–Murphy elements, revealing a deep structural parallel with random matrix theory. The framework enables precise asymptotic analysis of symmetric group characters and Plancherel measure on Young diagrams, offering exact formulas valid for all q, not just asymptotically.

ABSTRACT

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily and we show that it very closely related to the combinatorial approach to random matrices. Our formulas are exact (in a sense that they hold not only asymptotically for large q). This result has many interesting applications, for example it allows to find precise asymptotics of characters of large symmetric groups and asymptotics of the Plancherel measure on Young diagrams.

Motivation & Objective

• To develop a combinatorial framework that simplifies the convolution of conjugacy class indicators in symmetric groups.
• To analyze the structure of products of Jucys–Murphy element moments, which are otherwise highly complex and involve many conjugacy classes.
• To establish a precise, non-asymptotic connection between symmetric group combinatorics and random matrix theory.
• To enable exact asymptotic computations of characters of large symmetric groups and the Plancherel measure on Young diagrams.

Proposed method

• Introduces a combinatorial setup that models conjugacy class convolutions and Jucys–Murphy moment products using structured graph-like objects.
• Uses exact algebraic manipulation of symmetric group class functions to express complex combinations as manageable combinatorial sums.
• Leverages the known combinatorial machinery of random matrix theory—particularly moment calculations and diagram expansions—to inform and constrain the symmetric group constructions.
• Establishes a duality between the symmetric group’s conjugacy class algebra and the moment algebra of Jucys–Murphy elements via shared combinatorial patterns.
• Applies this duality to derive exact formulas that hold for all finite q, not only in the large q limit.
• Translates results from random matrix theory into precise statements about symmetric group representation theory and Young diagram measures.

Experimental results

Research questions

• RQ1 How can the convolution of two conjugacy class indicators in the symmetric group S_q be systematically simplified beyond asymptotic approximations?
• RQ2 What is the precise combinatorial structure underlying products of Jucys–Murphy element moments in S_q?
• RQ3 In what way do the combinatorics of symmetric group conjugacy classes mirror those of random matrix moments?
• RQ4 Can exact formulas for symmetric group characters and Plancherel measure asymptotics be derived using this framework?
• RQ5 What is the nature of the exact correspondence between symmetric group representation theory and random matrix theory in this setting?

Key findings

• The paper constructs an exact combinatorial correspondence between symmetric group conjugacy class convolutions and random matrix moment calculations, valid for all finite q.
• It demonstrates that products of Jucys–Murphy element moments and conjugacy class convolutions share identical combinatorial structures, enabling mutual transfer of results.
• The framework allows for the derivation of precise asymptotics of characters of large symmetric groups, going beyond standard asymptotic approximations.
• It provides exact formulas for the Plancherel measure on Young diagrams, including detailed asymptotic behavior in the large q limit.
• The method reveals that the complexity of conjugacy class products and moment expressions is governed by the same underlying combinatorial rules as in random matrix theory.
• The results are exact and not limited to large q, offering a new exact tool for studying symmetric group representation theory and its connections to random matrix ensembles.