Lakshmibai-Seshadri paths for hyperbolic Kac-Moody algebras of rank $2$

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $\lambda :={\Lambda }_1-{\Lambda }_2$, where ${\Lambda }_1,{\Lambda }_2$ are the fundamental weights for $\mathfrak{g}$; note that $\lambda$ is neither dominant nor antidominant. Let $\mathbb{B}(\lambda )$ be the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. We prove that (the crystal graph of) $\mathbb{B}(\lambda )$ is connected. Furthermore, we give an explicit description of Lakshmibai-Seshadri paths of shape $\lambda$.

arXiv comment15 pages