Jacopo Gandini (Bologna)
Abstract
Let G be a complex simple algebraic group and let X = G/K be an affine G-homogeneous variety whose coordinate ring C[X] is a multiplicity free G-module. Then X is called a multiplicity free G-variety, and (G,K) is called a reductive spherical pair. While the G-module structure of C[X] is well understood in terms of suitable combinatorial invariants associated to X, the G-algebra structure of C[X] is not so well understood. In the talk, I will consider the problem of decomposing the product of irreducible components in C[X] into irreducible summands.
A fundamental class of varieties which falls into the previous context is that of the symmetric varieties. Similarly to the restricted root system of a symmetric variety, also in the general case it is possibile to attach to X a root system which controls a lot of its geometry (and which is a main ingredient in the classification of these varieties). When the root system associated to X is of type A, I will propose a conjectural decomposition rule for the product of irreducible G-submodules of C[X], which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one symmetric case, I will also explain how this decomposition rule holds true whenever the root system associated to X is direct sum of subsystems of rank one.
The talk is based on a joint work with Paolo Bravi.