# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.10919