Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, Sphericity and Singularity Consequences

Reuven Hodges, Venkatramani Lakshmibai

Published 2016-09-29Version 1

Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N(\mathbb{C})$ (for left multiplication) of a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. For the natural action of $L_w$ on $\mathbb{C}[X(w)]$, the homogeneous coordinate ring of $X(w)$ (for the Pl\"ucker embedding), we give a combinatorial description of the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L_w$-modules; in fact, our description holds more generally for the action of the Levi part $L$ of any parabolic group $Q$ that is a subgroup of $Q_w$. Using this combinatorial description, we give a classification of all Schubert varieties $X(w)$ in the Grassmannian $G_{d,N}$ for which $\mathbb{C}[X(w)]$ has a decomposition into irreducible $L_w$-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical $L_w$-varieties. These classes include all smooth Schubert varieties, all determinantal Schubert varieties, as well as all Schubert varieties in $G_{2,N}$ and $G_{3,N}$. Also, as an important consequence, we get interesting results related to the singular locus of $X(w)$ and multiplicities at $T$-fixed points in $X(w)$.