Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$

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.