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Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients

Published online by Cambridge University Press:  20 November 2018

William Graham  and
Markus Hunziker
William Graham
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602-7403, USA, [email protected]
Markus Hunziker
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602-7403, USA, [email protected]
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Abstract

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Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda $ denotes the isomorphism class of an irreducible representation of $K$, let $\rho \lambda :K\to GL({{V}_{\lambda }})$ denote the corresponding irreducible representation and ${{S}_{\lambda }}$ the $\lambda $-isotypic component of $S$. Write ${{S}_{\lambda }}.{{S}_{\mu }}$ for the subspace of $S$ spanned by products of ${{S}_{\lambda }}$ and ${{S}_{\mu }}$. If ${{V}_{v}}$ occurs as an irreducible constituent of ${{V}_{\lambda }}\otimes {{V}_{\mu }}$, is it true that ${{S}_{v}}\subseteq {{S}_{\lambda }}.{{S}_{\mu }}?$ In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood–Richardson rule.

Keywords

14L3022E46Hermitian symmetric spacesmultiplicity free actionsLittlewood–Richardson coefficientsJack polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 2 , 01 April 2009 , pp. 351 - 372
Copyright
Copyright © Canadian Mathematical Society 2009

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