Adapted Sequence for Polyhedral Realization of Crystal Bases

TL;DR

This paper introduces the concept of adapted sequences to simplify the verification of positivity conditions in polyhedral realizations of crystal bases, providing explicit forms for classical Lie algebras.

Contribution

It defines adapted sequences and proves their sufficiency for positivity conditions, offering explicit polyhedral realizations for classical Lie algebras.

Findings

Adapted sequences ensure the positivity condition for classical Lie algebras.

Explicit polyhedral realizations are derived for arbitrary adapted sequences.

Simplifies the construction of crystal bases by reducing explicit calculations.

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.