Adapted Sequence for Polyhedral Realization of Crystal Bases

Abstract

The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base $B(\infty)$ as a polyhedral convex cone in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$. To construct the polyhedral realization, we need to fix an infinite sequence $\iota$ from the indices of the simple roots. According to this $\iota$, one has certain set of linear functions defining a polyhedral convex cone and under the `positivity condition' on $\iota$, it has been shown that the polyhedral convex cone is isomorphic to the crystal base $B(\infty)$. To confirm the positivity condition for a given $\iota$, 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 $\iota$ 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 $\iota$ in terms of column tableaux.