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Representation Theory

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Semisymmetric polynomials and the invariant theory of matrix vector pairs


Author: Friedrich Knop
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
Published electronically: August 15, 2001
MathSciNet review: 1857081
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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 $r=\frac12$ 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.


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Additional Information

Friedrich Knop
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: knop@math.rutgers.edu

DOI: https://doi.org/10.1090/S1088-4165-01-00129-7
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
Article copyright: © Copyright 2001 American Mathematical Society

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