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

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 $\Psi_{\iota}:B(\infty)\hookrightarrow \mathbb{Z}^{\infty}_{\iota}$, where $\iota$ is an infinite sequence of indices and $\mathbb{Z}^{\infty}_{\iota}$ 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 ${\rm Im}(\Psi_{\iota})$. In this article, supposing that $\mathfrak{g}$ is of affine type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$ or ${\rm D}^{(2)}_{n}$ and $\iota$ satisfies the condition of `adaptedness', we describe ${\rm Im}(\Psi_{\iota})$ by using several combinatorial objects such as extended Young diagrams and Young walls.