Alcove path model for $B(\infty)$

TL;DR

This paper develops an alcove path model for the crystal basis $B()$, connecting it to the Littelmann path model and its dual, providing a new combinatorial framework and insights into the structure of $B()$.

Contribution

It introduces a new alcove path model for $B()$ and establishes its connection to the Littelmann path model and its dual, expanding the combinatorial understanding of crystal bases.

Findings

The alcove path model's continuous limit recovers the dual Littelmann path model.

The dual alcove path model yields analogous results, linking to Li and Zhang's model.

The work provides a new combinatorial approach to studying $B()$.

Abstract

We construct a model for $B(\infty)$ using the alcove path model of Lenart and Postnikov. We show that the continuous limit of our model recovers a dual version of the Littelmann path model for $B(\infty)$ given by Li and Zhang. Furthermore, we consider the dual version of the alcove path model and obtain analogous results for the dual model, where the continuous limit gives the Li and Zhang model.