Discrete analogues of Macdonald-Mehta integrals

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.

50 pages

References in corpus (2)

• Multiple binomial sums
• Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles

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• Minor summation formula of hyperpfaffians and Selberg integrals