Polyhedral realizations for $B(\infty )$ and extended Young diagrams, Young walls of type ${\mathrm{A}}_{n-1}^{(1)}$, ${\mathrm{C}}_{n-1}^{(1)}$, ${\mathrm{A}}_{2n-2}^{(2)}$, ${\mathrm{D}}_n^{(2)}$

Abstract

The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras $U_q(\mathfrak{g})$. The polyhedral realization for $B(\infty )$ is a combinatorial description of the crystal base, which is defined as an image of embedding ${\Psi }_{\iota }:B(\infty )\hookrightarrow {\mathbb{Z}}_{\iota }^{\infty }$, where $\iota$ is an infinite sequence of indices and ${\mathbb{Z}}_{\iota }^{\infty }$ is an infinite $\mathbb{Z}$-lattice with a crystal structure associated with $\iota$. It is a natural problem to find an explicit form of the polyhedral realization $\mathrm{Im}({\Psi }_{\iota })$. In this article, supposing that $\mathfrak{g}$ is of affine type ${\mathrm{A}}_{n-1}^{(1)}$, ${\mathrm{C}}_{n-1}^{(1)}$, ${\mathrm{A}}_{2n-2}^{(2)}$ or ${\mathrm{D}}_n^{(2)}$ and $\iota$ satisfies the condition of `adaptedness', we describe $\mathrm{Im}({\Psi }_{\iota })$ by using several combinatorial objects such as extended Young diagrams and Young walls.