A classification of spherical Schubert varieties in the Grassmannian

TL;DR

This paper classifies spherical Schubert varieties in Grassmannians by analyzing their irreducible module decompositions under Levi subgroup actions, providing a comprehensive understanding of their structure.

Contribution

It offers a complete classification of spherical Schubert varieties in Grassmannians based on multiplicity-free decompositions of their coordinate rings.

Findings

Identified all multiplicity-free decompositions of coordinate rings.

Provided a classification of spherical Schubert varieties.

Connected module decompositions with geometric properties.

Abstract

Let $L$ be a Levi subgroup of $GL_N$ which acts by left multiplication on a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. We say that $X(w)$ is a spherical Schubert variety if $X(w)$ is a spherical variety for the action of $L$. In earlier work we provide a combinatorial description of the decomposition of the homogeneous coordinate ring of $X(w)$ into irreducible $L$-modules for the induced action of $L$. In this work we classify those decompositions into irreducible $L$-modules that are multiplicity-free. This is then applied towards giving a complete classification of the spherical Schubert varieties in the Grassmannian.