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 $B(\infty )$ 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 ι 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 ι, and 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 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.

Keywords:

• Column tableaux
• crystal bases
• polyhedral realizations

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B37
• 81R50

Additional information

Funding

Y.K. is supported by JSPS KAKENHI Grant Number JP17H07103, and T.N. is supported in part by JSPS KAKENHI Grant Number JP15K04794.