Introduction

The adjoining of sign changes to the symmetric group produces the hyperoctahedral group. Many techniques and results from the previous chapter can be adapted to this group by considering only functions that are even in each variable. A second parameter κ′ is associated with the conjugacy class of sign changes. The main part of the chapter begins with a description of the differential–difference operators for these groups and their effect on polynomials of arbitrary parity (odd in some variables, even in the others). As in the type-A case there is a fundamental set of first-order commuting self-adjoint operators, and their eigenfunctions are expressed in terms of nonsymmetric Jack polynomials. The normalizing constant for the Hermite polynomials, that is, the Macdonald–Mehta–Selberg integral, is computed by the use of a recurrence relation and analytic-function techniques. There is a generalization of binomial coefficients for the nonsymmetric Jack polynomials which can be used for the calculation of the Hermite polynomials. Although no closed form is as yet available for these coefficients, we present an algorithmic scheme for obtaining specific desired values (by symbolic computation). Calogero and Sutherland were the first to study nontrivial examples of many-body quantum models and to show their complete integrability.