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

TL;DR

This paper provides explicit polyhedral realizations of crystal bases for certain affine Lie algebras using combinatorial objects like Young diagrams and walls, enhancing understanding of their structure.

Contribution

It explicitly describes the polyhedral realization of $B()$ for affine types ${ m A}^{(1)}_{n-1}$, ${ m C}^{(1)}_{n-1}$, ${ m A}^{(2)}_{2n-2}$, ${ m D}^{(2)}_{n}$ using combinatorial models.

Findings

Explicit description of the image of the crystal embedding

Use of extended Young diagrams and Young walls

Applicable to specific affine types

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.