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

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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 \(\textrm{A}^{(1)}_{n-1}\), \(\textrm{C}^{(1)}_{n-1}\), \(\textrm{A}^{(2)}_{2n-2}\) or \(\textrm{D}^{(2)}_{n}\). 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.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP20J00186.

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  1. Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan

    Yuki Kanakubo

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Kanakubo, Y. Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type \(\textrm{A}^{(1)}_{n-1}\), \(\textrm{C}^{(1)}_{n-1}\), \(\textrm{A}^{(2)}_{2n-2}\), \(\textrm{D}^{(2)}_{n}\). Lett Math Phys 113, 60 (2023). https://doi.org/10.1007/s11005-023-01680-0

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