Combinatorial formula for Macdonald polynomials, Bethe Ansatz, and generic Macdonald polynomials

摘要

We give a direct proof of the combinatorial formula for interpolationMacdonald polynomials by introducing certain polynomials, which we call genericMacdonald polynomials, which depend on $d$ additional parameters and specializeto all Macdonald polynomials of degree $d$. The form of these genericpolynomials is that of a Bethe eigenfunction and they imitate, on a moreelementary level, the $R$-matrix construction of quantum immanants.