Degenerate coordinate rings of flag varieties and Frobenius splitting

TL;DR

This paper constructs Frobenius split degenerations of coordinate rings of classical flag varieties across all types, providing new proofs and extending known results to include the G2 type.

Contribution

It introduces a new degeneration of coordinate rings for classical flag varieties in all types and proves Frobenius splitting in most cases, including the G2 type.

Findings

Frobenius splitting of degenerate flag varieties in types A, C, and G2.

An alternative proof of Frobenius splitting for types A and C.

A representation-theoretic condition related to Frobenius splitting of classical flag varieties.

Abstract

Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an alternate proof of splitting in types $A_n$ and $C_n$; the case $G_2$ was not previously known. We also give a representation-theoretic condition on PBW-graded versions of Weyl modules which is equivalent to the existence of a Frobenius splitting of the classical flag variety that maximally compatibly splits the identity.