A geometric approach to Standard Monomial Theory

TL;DR

This paper introduces a geometric method to construct a standard monomial basis for coordinate rings of flag varieties, compatible with Schubert structures, using vanishing theorems and degeneration techniques.

Contribution

It provides a new geometric construction of standard monomial bases applicable to all flag varieties and their coordinate rings, including classical types.

Findings

Constructed a standard monomial basis compatible with Schubert varieties.

Applied vanishing theorems and diagonal degeneration techniques.

Extended results to multi-homogeneous coordinate rings of classical type.

Abstract

We obtain a geometric construction of a ``standard monomial basis'' for the homogeneous coordinate ring associated with any ample line bundle on any flag variety. This basis is compatible with Schubert varieties, opposite Schubert varieties, and unions of intersections of these varieties. Our approach relies on vanishing theorems and a degeneration of the diagonal ; it also yields a standard monomial basis for the multi-homogeneous coordinate rings of flag varieties of classical type.