Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence

TL;DR

This paper studies marked poset polytopes, establishing conditions for their unimodular equivalence, analyzing their Minkowski sum structure, and decomposing them into indecomposable components, with implications for representation theory.

Contribution

It generalizes known results on marked poset polytopes, characterizes unimodular equivalence, and introduces a Minkowski sum and indecomposable decomposition framework.

Findings

Marked chain and order polytopes are unimodular equivalent under certain conditions.

Lattice points in marked poset polytopes are Minkowski sums of 0-1 polytope lattice points.

Marked posets can be decomposed into indecomposable components respecting Minkowski sums.

Abstract

We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes. Moreover, we provide a decomposition of the marked poset into indecomposable marked posets, which respects this Minkowski sum decomposition for the marked chain polytopes polytopes.