CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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

  Published:2009-04-01
 Printed: Apr 2009
  • William Graham
  • Markus Hunziker
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

Abstract

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 $\rho_\lambda\from K \rightarrow GL(V_{\lambda})$ denote the corresponding irreducible representation and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write $S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of $S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible constituent of $V_\lambda\otimes V_\mu$, is it true that $S_\nu\subseteq S_\lambda\cdot S_\mu$? 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: Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials
MSC Classifications: 14L30, 22E46 show english descriptions Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Semisimple Lie groups and their representations
14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
22E46 - Semisimple Lie groups and their representations
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/