Standard monomial theory and toric degenerations of Richardson varieties in flag varieties

TL;DR

This paper develops a combinatorial method for constructing standard monomial bases for certain Richardson varieties in flag varieties and applies it to produce new toric degenerations, advancing understanding of their algebraic and geometric structures.

Contribution

It introduces a direct combinatorial rule for standard monomial bases for a family of Richardson varieties and uses this to generate new toric degenerations.

Findings

Provided a straightforward combinatorial rule for standard monomial bases.

Constructed a new family of toric degenerations for Richardson varieties.

Enhanced understanding of the algebraic structure of Richardson varieties.

Abstract

We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richardson varieties inside flag varieties.