LS Algebras, Valuations and Schubert Varieties

TL;DR

This paper introduces LS algebras as a new algebraic framework connecting standard monomial theory, Newton-Okounkov bodies, and valuations, specifically applied to Schubert varieties, revealing how LS paths encode vanishing multiplicities.

Contribution

It develops an algebraic approach using LS algebras to relate monomial theories, valuations, and geometric structures of Schubert varieties, extending previous stratification methods.

Findings

LS paths encode vanishing multiplicities in Schubert varieties

The approach aligns with existing Seshadri stratification methods

Provides a new algebraic perspective linking combinatorics and geometry

Abstract

In this paper, we propose an algebraic approach via Lakshmibai-Seshadri (LS) algebras to establish a link between standard monomial theories, Newton-Okounkov bodies and valuations. This is applied to Schubert varieties, where this approach is compatible with the one using Seshadri stratifications by the same authors (arXiv:2112.03776), showing that LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.