## 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.## References

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## Bibliographic Information

**Friedrich Knop**- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- MR Author ID: 103390
- ORCID: 0000-0002-4908-4060
- Email: knop@math.rutgers.edu
- Received by editor(s): October 14, 1999
- Received by editor(s) in revised form: May 12, 2001
- Published electronically: August 15, 2001
- Additional Notes: This work was partially supported by a grant of the NSF
- © Copyright 2001 American Mathematical Society
- Journal: Represent. Theory
**5**(2001), 224-266 - MSC (2000): Primary 33D55, 20G05, 39A70, 05E35
- DOI: https://doi.org/10.1090/S1088-4165-01-00129-7
- MathSciNet review: 1857081