Pieri Rules for the Jack Polynomials in Superspace and the 6-Vertex Model

Abstract

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in α (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.Key words and phrases. Jack polynomials, Pieri rules, symmetric functions in superspace, 6-vertex model. 1 We should mention that special cases were considered in [7].

The norm and the Evaluation of the Macdonald polynomials in superspace

“…Most of the important features of the Macdonald polynomials seem to extend to superspace. The duality and the existence of Macdonald operators were shown for instance in [5] while various properties were conjectured to hold such as explicit formulas for the norm and the evaluation [4], Pieri rules (which connect to the 6-vertex model) [9] and symmetry [3]. Most remarkably, an extension of the original Macdonald positivity conjecture was stated in [5].…”