Semisymmetric polynomials and the invariant theory of matrix vector pairs

Abstract

In this paper we introduce and investigate a one-parameter family of polynomials. They are semisymmetric, i.e. symmetric in the variables with odd and even index separately. In fact, the family forms a basis of the space of semisymmetric polynomials. For two values of the parameter r, namely r=1/2 and r=1, the polynomials have a representation theoretic meaning. In general, they form the semisymmetric analogue of (shifted) Jack polynomials.