Discrete analogues of Macdonald-Mehta integrals

TL;DR

This paper develops discrete analogues of Macdonald-Mehta integrals for classical reflection groups, providing closed-form evaluations for specific cases and deriving new formulas related to group characters and elliptic hypergeometric series.

Contribution

It introduces novel discrete versions of Macdonald-Mehta integrals for classical groups and derives explicit formulas using character identities and hypergeometric transformations.

Findings

Closed-form evaluations for exponent-1 cases using character identities.

Transformation formulas for elliptic hypergeometric series for exponent-2 cases.

New product formulas for counting orthogonal and symplectic tableaux.

Abstract

We consider discretisations of the Macdonald--Mehta integrals from the theory of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$, $\mathrm{B}_r$ and $\mathrm{D}_r$, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents $1$ and $2$. Our proofs for the exponent-$1$ cases rely on identities for classical group characters, while most of the formulas for the exponent-$2$ cases are derived from a transformation formula for elliptic hypergeometric series for the root system $\mathrm{BC}_r$. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.