From superintegrability to tridiagonal representation of $\beta$-ensembles

TL;DR

This paper connects superintegrability of Jack polynomial averages with the tridiagonal matrix representation of $eta$-ensembles, providing a clearer understanding and formal proof via Virasoro constraints, and discusses symmetry breaking phenomena.

Contribution

It reveals the compatibility of superintegrability with the tridiagonal matrix representation of $eta$-ensembles, offering a transparent interpretation and formal proof using Virasoro constraints.

Findings

Superintegrability aligns with the tridiagonal representation.

Virasoro constraints provide a formal proof.

Spontaneous symmetry breaking observed in the measure.

Abstract

The wonderful formulas by I.Dumitriu and A.Edelman rewrite $\beta$-ensemble, with eigenvalue integrals containing Vandermonde factors in the power $2\beta$, through integrals over tridiagonal matrices, where $\beta$-dependent are the powers of individual matrix elements, not their differences. These potentially useful formulas are usually deduced from rather complicated and non-transparent combinatorics and are not as widely known as they deserve. We explain that the superintegrability property, i.e. a simple expression of the Gaussian averages of arbitrary Jack polynomials through the same Jack polynomials, is immediately consistent with this tridiagonal representation, which may serve as a clue to its simple and transparent interpretation. For a formal non-perturbative proof, we use the Virasoro constraints, which themselves acquire an interesting structure in the tridiagonal realization. We also attract attention to the surprising spontaneous breakdown of discrete invariance by the tridiagonal measure, which may signal a new interesting anomaly at the elementary level of the basic eigenvalue matrix model.