Pieri-Type Formulas for Nonsymmetric Macdonald Polynomials

Abstract

In symmetric Macdonald polynomial theory, the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function e r (z) and the Macdonald polynomial $P_{\kappa}(z)$ . In this paper we give the nonsymmetric analogs for the cases r=1 and r= n−1. We do this by first deducing the decomposition for the product of any nonsymmetric Macdonald polynomial $E_{\eta}(z)$ with z i in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of $e_1(z)E_{\eta}(z)$ we evaluate the generalized binomial coefficients $(^{\eta}_v)$ associated with the nonsymmetric Macdonald polynomials for $|\eta|=|v|+1$ .