Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

TL;DR

This paper characterizes when Mirkovic-Vilonen (MV) polytopes in highest weight crystals belong to specific Demazure or opposite Demazure crystals using edge length conditions related to Weyl group elements.

Contribution

It provides a necessary and sufficient condition based on edge lengths for MV polytopes to lie in Demazure crystals, extending understanding of their combinatorial structure.

Findings

Edge length conditions determine Demazure crystal membership.

Explicit description of extremal MV polytopes as pseudo-Weyl polytopes.

Polytopal criteria for MV polytopes in opposite Demazure crystals.

Abstract

We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of $P$ corresponding to a reduced word for the longest element of the Weyl group $W$. % Also, we give an explicit description as a pseudo-Weyl polytope for extremal MV polytopes in a highest weight crystal. % Finally, by combining the results above, we obtain a polytopal condition for an MV polytope $P$ to lie in an arbitrary fixed opposite Demazure crystal.