Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects 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 develops explicit polyhedral models for crystal bases of certain affine Lie algebra modules using combinatorial objects like Young diagrams, providing new insights into their structure and associated functions.

Contribution

It introduces explicit polyhedral realizations of crystal bases for classical affine Lie algebra modules using combinatorial objects, extending prior models.

Findings

Explicit polyhedral forms for crystal bases are given.

A combinatorial description of epsilon functions on B(infinity) is provided.

Connections between Young diagrams and crystal structures are established.

Abstract

In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$ or ${\rm D}^{(2)}_{n}$. We will give explicit forms of polyhedral realizations in terms of extended Young diagrams or Young walls that appear in the representation theory of quantized enveloping algebras of classical affine type. As an application, a combinatorial description of $\varepsilon_k^*$ functions on $B(\infty)$ will be given.