Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)

TL;DR

This paper investigates the autocorrelation functions of ratios of random characteristic polynomials for unitary groups, utilizing supersymmetry and Howe pairs to derive explicit character formulas applicable across all matrix sizes.

Contribution

It introduces a novel approach combining Howe duality and superanalysis to compute autocorrelation functions for all N, extending previous stable-range results to the full range of matrix dimensions.

Findings

Derived explicit character formulas for ratios of characteristic polynomials.

Extended previous stable-range results to all matrix sizes N.

Connected superalgebra representations with classical Lie group characters.

Abstract

For the classical compact Lie groups K = U(N) the autocorrelation functions of ratios of random characteristic polynomials are studied. Basic to our treatment is a property shared by the spinor representation of the spin group with the Shale-Weil representation of the metaplectic group: in both cases the character is the analytic square root of a determinant or the reciprocal thereof. By combining this fact with Howe's theory of supersymmetric dual pairs (g,K), we express the K-Haar average product of p ratios of characteristic polynomials and q conjugate ratios as a character which is associated with an irreducible representation of the Lie superalgebra g = gl(n|n) for n = p+q. This primitive character is shown to extend to an analytic radial section of a real supermanifold related to gl(n|n), and is computed by invoking Berezin's description of the radial parts of Laplace-Casimir operators for gl(n|n). The final result for the character looks like a natural transcription of the Weyl character formula to the context of highest-weight representations of Lie supergroups. While several other works have recently reproduced our results in the stable range where N is no less than max(p,q), the present approach covers the full range of matrix dimensions N. To make this paper accessible to the non-expert reader, we have included a chapter containing the required background material from superanalysis.