Lusztig polytopes and FFLV polytopes

TL;DR

This paper proves that in type A_n, the FFLV polytope equals the Minkowski sum of Lusztig polytopes from different decompositions and proposes a conjecture on crystal structures.

Contribution

It establishes the equivalence of FFLV and Minkowski sums of Lusztig polytopes in type A_n and introduces a conjecture on their crystal structures.

Findings

FFLV polytope equals Minkowski sum of Lusztig polytopes in type A_n

Formulation of a conjecture on crystal structures of FFLV polytopes

Insight into polytope decompositions and crystal combinatorics

Abstract

In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a conjecture about the crystal structures on FFLV polytopes.