Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 19 (2009), No. 4, 771--795Copyright 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 [ Fulltext-pdf  (259  KB)] for subscribers only.