Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type \(\mathrm{A}^{(1)}_{n-1}, \mathrm{C}^{(1)}_{n-1}, \mathrm{A}^{(2)}_{2n-2}, \mathrm{D}^{(2)}_n\) (Q6113200)

In the paper under review, the author investigates the polyhedral realization of the integrable highest weight crystal \(B(\lambda)\) of affine type \(A_{n-1}^{(1)}\), \(C_{n-1}^{(1)}\), \(A_{2n-2}^{(2)}\) and \(D_{n}^{(2)}\). Let \(\Psi_{\iota}^{(\lambda)} : B(\lambda) \longrightarrow \mathbb{Z}_\iota^\infty \otimes R_\lambda\) be the crystal embedding in the polyhedral realization setting. Under the condition that \(\iota\) is adapted, the author gives an explicit description of the image \( \mathrm{Im} \Psi_{\iota}^{(\lambda)}\) in terms of extended Young diagrams and Young walls from the viewpoint of combinatorics. This is a continuation of the author's previous work on the polyhedral realization of the infinite crystal \(B(\infty)\) [\textit{Y. Kanakubo}, Algebr. Represent. Theory 26, No. 5, 2181--2233 (2023; Zbl 1527.17010)]. As an application, the author gives a combinatorial description of \(\varepsilon_i^*\) on the infinite crystal \(B(\infty)\).