Journal of Lie Theory 19 (2009), No. 4, 771--795
Copyright Heldermann Verlag 2009
Invariant Polynomials for Multiplicity Free Actions
Chal Benson
Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
bensonf@ecu.edu
R. Michael Howe
Dept. of Mathematics, University of Wisconsin, Eau Claire, WI 54701-4004, U.S.A.
hower@uwec.edu
Gail Ratcliff
Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
ratcliffg@ecu.edu
[Abstract-pdf]
\def\C{{\Bbb C}} \def\R{{\Bbb R}} \def\HH{{\Bbb H}} This work concerns linear multiplicity free actions of the complex groups $G_\C=GL(n,\C)$, $GL(n,\C)\times GL(n,\C)$ and $GL(2n,\C)$ on the vector spaces $V=Sym(n,\C)$, $M_n(\C)$ and $Skew(2n,\C)$. We relate the canonical invariants in $\C[V \oplus V^*]$ to spherical functions for Riemannian symmetric pairs $(G,K)$ where $G=GL(n,\R)$, $GL(n,\C)$ or $GL(n,\HH)$ respectively. These in turn can be expressed using three families of classical zonal polynomials. We use this fact to derive a combinatorial algorithm for the generalized binomial coefficients in each case. Many of these results were obtained previously by Knop and Sahi using different methods.
Keywords: Multiplicity free actions, invariant theory, symmetric functions.
MSC: 20G05, 13A50; 05E15
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