Multiprojective Seshadri stratifications for Schubert varieties and standard monomial theory

TL;DR

This paper introduces a geometric framework for understanding standard monomial bases of Schubert varieties using Seshadri stratifications, generalizing classical tableaux to all Dynkin types.

Contribution

It provides a new geometric interpretation of Lakshmibai-Seshadri tableaux and constructs filtrations of coordinate rings with subquotients indexed by these tableaux.

Findings

Filtrations of coordinate rings are constructed with one-dimensional subquotients.

The approach generalizes classical tableaux to all Dynkin types.

Provides a geometric perspective on standard monomial theory.

Abstract

Using the language of Seshadri stratifications we develop a geometrical interpretation of Lakshmibai-Seshadri-tableaux and their associated standard monomial bases. These tableaux are a generalization of Young-tableaux and De-Concini-tableaux to all Dynkin types. More precisely, we construct filtrations of multihomogeneous coordinate rings of Schubert varieties, such that the subquotients are one-dimensional and indexed by standard tableaux.