Following Schubert varieties under Feigin's degeneration of the flag variety

TL;DR

This paper investigates how Feigin's flat degeneration affects Schubert varieties in type A flag varieties, revealing conditions for irreducibility and linking degenerations to symmetric group combinatorics.

Contribution

It provides a detailed description of the degeneration's impact on Schubert varieties and connects reducibility to combinatorial properties, also identifying some as Richardson varieties.

Findings

Conditions for irreducibility of degenerated Schubert varieties

Combinatorial criteria for reducibility based on symmetric group

Identification of certain Schubert varieties with Richardson varieties

Abstract

We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.