Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations

TL;DR

This paper constructs and analyzes $eta$-deformed partition function hierarchies with $W$-representations, demonstrating superintegrability and deriving character expansions, encompassing several well-known matrix models.

Contribution

It introduces a unified framework for $eta$-deformed partition functions with $W$-representations, revealing superintegrability and character expansions for various matrix models.

Findings

Superintegrability property established for the hierarchies.

Character expansions derived for Schur functions and Jack polynomials.

Contains several well-known matrix models within the hierarchy.

Abstract

We construct the ($\beta$-deformed) partition function hierarchies with $W$-representations. Based on the $W$-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), $N\times N$ complex matrix model, $\beta$-deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.