Satoshi Naito and Daisuke Sagaki, Mirkovi\'c-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

We give a condition for an MV (Mirkovi\'c-Vilonen) polytope in a highest weight crystal to lie in a Demazure crystal or an opposite Demazure crystal in terms of the lengths (Lusztig datum) of edges along a path through the 1-skeleton of the polytope corresponding to a reduced word for the longest element of the Weyl group. Also, we give an explicit description of the initial and final directions of an LS (Lakshmibai-Seshadri) path in terms of MV polytopes under the correspondence (which is not yet explicit) between LS paths and MV polytopes.