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We study algebraic, combinatorial and geometric aspects of weighted PBW-typedegenerations of (partial) flag varieties in type $A$. These degenerations arelabeled 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 classicalflag variety, its abelian PBW degeneration, some of its linear degenerationsand a particular toric degeneration.

Weighted PBW degenerations and tropical flag varieties