Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

TL;DR

This paper explores toric degenerations of Schubert varieties within flag varieties, using combinatorial methods involving matching fields and tableaux to identify conditions for toric ideals.

Contribution

It introduces an analogue of matching field ideals for Schubert varieties and characterizes when these lead to toric degenerations, advancing combinatorial understanding.

Findings

Block diagonal matching fields induce toric degenerations.

Complete characterization of toric ideals among Schubert varieties.

Use of permutation combinatorics to analyze degenerations.

Abstract

We study Gr\"obner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial theory to show that block diagonal matching fields give rise to toric degenerations. Our methods and results use the combinatorics of permutations associated to Schubert varieties, matching fields and their corresponding tableaux.