Journal of Lie Theory

no. 4
Invariant Polynomials for Multiplicity Free Actions
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 
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     author = {C. Benson and R. M. Howe and G. Ratcliff},
     title = {Invariant {Polynomials} for {Multiplicity} {Free} {Actions}},
     journal = {Journal of Lie Theory},
     pages = {771--795},
     year = {2009},
     volume = {19},
     number = {4},
     doi = {10.5802/jolt.580},
     zbl = {1220.43004},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.580/}
}
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C. Benson; R. M. Howe; G. 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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