Semisymmetric polynomials and the invariant theory of matrix vector pairs

Knop, Friedrich

doi:10.1090/S1088-4165-01-00129-7

Semisymmetric polynomials and the invariant theory of matrix vector pairs
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by Friedrich Knop

Represent. Theory 5 (2001), 224-266

DOI: https://doi.org/10.1090/S1088-4165-01-00129-7

Published electronically: August 15, 2001

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Abstract:

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 $r=\frac 12$ 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.

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