Sum rules for characters from character-preservation property of matrix models

TL;DR

This paper explores the character-preservation property in matrix models, deriving sum rules and implications for $W$-representations, with a focus on theoretical understanding and combinatorial identities.

Contribution

It introduces new sum rules for characters in matrix models and connects these to $W$-representations and combinatorial identities involving single-hook diagrams.

Findings

Eigenvalue matrix models preserve character averages.

Explicit formulas for partition functions facilitate calculations.

Sum rules relate to $W$-representations and involve single-hook diagrams.

Abstract

One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame the remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-$N$ limit implies interesting combinatorial sum rules for characters. We describe also implications to $W$-representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities.