Reuven Hodges, Geometric Methods in Representation Theory

Title:   Levi subgroup actions on Schubert varieties in the Grassmannian

Abstract: Let L be the Levi part of the stabilizer in GL_N(C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian. For the induced action of L on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plucker embedding), I will give a combinatorial description of the decomposition of C[X(w)] into irreducible L-modules. Using this combinatorial description, I give a classification of all Schubert varieties X(w) in the Grassmannian for which C[X(w)] has a decomposition into irreducible L-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical L-varieties. Also, I will describe interesting related results on the singular locus of X(w) and multiplicities at points in X(w).