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.
Categories: math.CO math-ph math.MP