Journal of Lie Theory

no. 4
Invariant Polynomials for Multiplicity Free Actions
Chal Benson1; R. Michael Howe2; Gail Ratcliff1
1Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
2Dept. of Mathematics, University of Wisconsin, Eau Claire, WI 54701-4004, U.S.A.
Journal of Lie Theory, Volume 19 (2009) no. 4, pp. 771-795
\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.
DOI: 10.5802/jolt.580
Classification: 20G05, 13A50, 05E15
Keywords: Multiplicity free actions, invariant theory, symmetric functions

Chal Benson  1 ; R. Michael Howe  2 ; Gail Ratcliff  1

1 Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
2 Dept. of Mathematics, University of Wisconsin, Eau Claire, WI 54701-4004, U.S.A. 
Chal Benson; R. Michael Howe; Gail Ratcliff. Invariant Polynomials for Multiplicity Free Actions. Journal of Lie Theory, Volume 19 (2009) no. 4, pp. 771-795. doi: 10.5802/jolt.580
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     title = {Invariant {Polynomials} for {Multiplicity} {Free} {Actions}},
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     zbl = {1220.43004},
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