Monomial realizations and LS paths of fundamental representations for rank $2$ Kac-Moody algebras

TL;DR

This paper establishes an explicit isomorphism between Lakshmibai-Seshadri paths and monomial realizations for rank 2 Kac-Moody algebras, providing concrete descriptions and applications in terms of Weyl groups.

Contribution

It explicitly constructs an isomorphism between LS paths and monomial realizations for rank 2 Kac-Moody algebras, including a detailed form in terms of Weyl groups.

Findings

Explicit isomorphism between LS paths and monomial realizations.

Concrete description of monomial realization using Weyl groups.

Application to understanding fundamental representations.

Abstract

For a Kac-Moody algebra $\mathfrak{g}$ of rank $2$ and a fundamental weight $\lambda$, we explicitly give an isomorphism between the set of Lakshmibai-Seshadri paths $\mathbb{B}(\lambda)$ and monomial realization $\mathcal{M}(\lambda)$. As an application, we also give an explicit form of monomial realization $\mathcal{M}(\lambda)$ in terms of Weyl groups.