WKB-expansion of the HarishChandra-Itzykson-Zuber integral for arbitrary beta

TL;DR

This paper develops a WKB-expansion for the Harish-Chandra-Itzykson-Zuber integral across arbitrary beta values, extending previous results for classical groups and employing duality and symmetric function transformations.

Contribution

It introduces a generalized WKB-expansion for the Harish-Chandra-Itzykson-Zuber integral applicable to all beta values, including non-unitary symmetries, using duality and symmetric function techniques.

Findings

Derived a WKB-expansion from the heat kernel equation for general beta.

Connected zonal polynomial expansion to tau polynomial expansion for beta=1.

Provided a duality relation and transformation for symmetric functions in the context of the integral.

Abstract

This article is devoted to the asymptotic expansion of the generalized Harish Chandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterized by a parameter beta(as usual beta =1,2 and 4 correspond to the orthogonal, unitary and symplectic group integrals). A WKB-expansion for f is derived from the heat kernel differential equation, for general values of k and beta. From an expansion in terms of zonal polynomials, one obtain an expansion in powers of the tau's for beta=1, and generalizations are considered for general beta. A duality relation, and a transformation of products of pairs of symmetric functions into tau polynomials, is used to obtain the expression for f(tau ij) for general beta.