Tropical symplectic flag varieties: a Lie-theoretic approach

TL;DR

This paper explores the tropicalization of symplectic flag varieties, identifying a maximal prime cone whose associated degenerations relate to FFLV polytopes, using birational sequences to connect Lie algebra representations with algebraic geometry.

Contribution

It explicitly characterizes a maximal cone in the tropicalization of symplectic flag varieties and links it to toric degenerations via FFLV polytopes using birational sequences.

Findings

Identified a maximal prime cone in the tropicalization.

Connected Gr"obner degenerations to FFLV polytopes.

Established a bridge between Lie algebra representations and algebraic geometry.

Abstract

We study tropicalization of symplectic flag varieties with respect to the Pl\"ucker embedding. We identify a particular maximal prime cone in this tropicalization by explicitly giving its facets. For every interior point of this maximal cone, the corresponding Gr\"obner degeneration is the toric variety associated to the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope. Our main tool is the notion of birational sequences introduced by Fourier, Littelmann and the second author, which bridges between weighted PBW filtrations of representations of symplectic Lie algebras and degree functions on defining ideals of symplectic flag varieties.