On the B-canonical splittings of flag varieties

TL;DR

This paper characterizes which subvarieties of flag varieties are compatibly split by the canonical Frobenius splitting, showing they are precisely Richardson varieties, thus linking geometric splitting properties with combinatorial structures.

Contribution

It provides a complete characterization of compatibly split subvarieties of flag varieties in positive characteristic, identifying them as Richardson varieties.

Findings

Irreducible closed subvarieties compatibly split by the canonical Frobenius splitting are exactly Richardson varieties.

The result connects Frobenius splitting with the combinatorial structure of Schubert and Richardson varieties.

The paper establishes a criterion for compatibility with the canonical Frobenius splitting in flag varieties.

Abstract

Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.