Abstract
AbstractLetGbe a simple complex algebraic group, and let$K \subset G$be a reductive subgroup such that the coordinate ring of$G/K$is a multiplicity-freeG-module. We consider theG-algebra structure of$\mathbb C[G/K]$and study the decomposition into irreducible summands of the product of irreducibleG-submodules in$\mathbb C[G/K]$. When the spherical roots of$G/K$generate a root system of type$\mathsf A$, we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of$G/K$is a direct sum of subsystems of rank 1.
Highlights
Let G be a connected semisimple complex algebraic group, and let K ⊂ G be a reductive subgroup
If this is the case, we will say that K is a spherical subgroup, and that (G, K ) is a reductive spherical pair
By a result of Vinberg and Kimelfeld [34], reductive spherical subgroups are characterized by the property that the coordinate ring C[G/K] is a multiplicity free G-module
Summary
Let G be a connected semisimple complex algebraic group, and let K ⊂ G be a reductive subgroup. Our main contribution with the present paper is to show how a general conjectural rule for decomposing the product of two irreducible constituents in C[X] follows from Stanley’s mentioned conjecture for all the non-symmetric affine spherical homogeneous varieties whose spherical roots generate a root system of type A. When this root system is a direct sum of subsystems of rank one, which happens in most cases, we will see (with the exception of one case) that such description holds true thanks to Stanley’s Pieri rule for Jack symmetric functions. Levi subgroups (that is, to the case of the symmetric varieties of Hermitian type), and we formulate a conjectural rule for the decomposition of the product of two irreducible components in C[G/K] in this case as well (see Conjecture 6.2)
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