Two parameters circular ensembles and Jacobi-Trudi type formulas for Jack functions of rectangular shapes

Jack function theory is useful for the calculation of the moment of the characteristic polynomials in Dyson’s circular β-ensembles (CβE). We define a q-analogue of the CβE and calculate moments of characteristic polynomials via Macdonald function theory. By this q-deformation, the asymptotics calculation of these moments becomes simple and the ordinary CβE case is recovered as q → 1. Further, by using a hyperdeterminant which is a simple generalization of a determinant, we give a Jacobi-Trudi type formula for Jack symmetric functions of rectangular shapes. MSC-class: primary 15A52; secondary 05E05.