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

TL;DR

This paper provides a combinatorial description of how Levi subgroups act on the coordinate rings of Schubert varieties, revealing sphericity properties for various classes of these varieties.

Contribution

It introduces a new combinatorial framework for decomposing coordinate rings under Levi subgroup actions and establishes sphericity for several classes of Schubert varieties.

Findings

Decomposition of coordinate rings into irreducible modules described combinatorially

All smooth and determinantal Schubert varieties are spherical under Levi actions

Schubert varieties in G_{2,N} are spherical under Levi actions

Abstract

Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N$ (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 subgroup $Q$ that is contained in $Q_w$. This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in $G_{2,N}$ are spherical $L_w$-varieties.