Adapted sequence for polyhedral realization of crystal bases

Abstract

The polyhedral realization of crystal base has been introduced by Zelevinsky and Nakashima, which describe the crystal base as a polyhedral convex cone in the infinite -lattice To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. According to this one has certain set of linear functions defining a polyhedral convex cone and under the “positivity condition” on ι, and it has been shown that the polyhedral convex cone is isomorphic to the crystal base To confirm the positivity condition for a given ι, we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if ι is an adapted sequence, then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences ι in terms of column tableaux.