Keys and Evacuation via Virtualization

Abstract

In this paper, we study the relation between the key map and virtualization of crystals. Namely, we prove that virtualization between crystals in any two finite Cartan types commutes with the left and right key maps, thus embedding Demazure crystals and atoms correspondingly. In particular, this implies that the key map in any finite Cartan type can be reduced to the key map in a simply-laced type, provided an appropriate virtualization exists, generalizing the work of Azenhas--Santos. As an application, we study these maps in the context of orthogonal Kashiwara--Nakashima tableaux and show that the virtualizations from type B into C considered independently by Fujita and Pappe--Pfannerer--Schilling--Simone coincide with the splitting map of De Concini and Lecouvey. As a consequence, this enables us to give a new and purely combinatorial definition of orthogonal evacuation.