
Journal of Lie Theory 19 (2009), No. 4, 771795 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 547014004, U.S.A. hower@uwec.edu Gail Ratcliff Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A. ratcliffg@ecu.edu [Abstractpdf] \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 [ Fulltextpdf (259 KB)] for subscribers only. 