Semisymmetric polynomials and the invariant theory of matrix vector pairs

We introduce and investigate a one-parameter family of multivariate polynomials $R_\lambda$. They form a basis of the space of semisymmetric polynomials, i.e., those polynomials which are symmetric in the variables with odd and even index separately. For two values of the parameter $r$, namely $r=\frac12$ and $r=1$, the polynomials have a representation theoretic meaning related to matrix-vector pairs. In general, they form the semisymmetric analogue of (shifted) Jack polynomials. Our main result is that the $R_\lambda$ are joint eigenfunctions of certain difference operators. From this we deduce, among others, the Extra Vanishing Theorem, Triangularity, and Pieri Formulas.