Weighted PBW degenerations and tropical flag varieties

TL;DR

This paper explores algebraic, combinatorial, and geometric properties of weighted PBW degenerations of flag varieties, linking them to tropical geometry and revealing various classical and degenerate structures within a unified framework.

Contribution

It introduces a polyhedral cone parameterizing weighted PBW degenerations, connecting these degenerations to tropical flag varieties and unifying several classical and degenerate cases.

Findings

Degenerations correspond to points in a polyhedral cone within the tropical flag variety

Classical flag variety, abelian PBW degeneration, and toric degenerations are special cases within this framework

The structure of the degenerations can be explicitly described via degree functions in the cone

Abstract

We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type $A$. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.