Nakajima monomials, Young walls and Kashiwara embedding for \(U_q(A_n^{(1)})\)
In this paper, the authors investigate three kinds of realizations of the level 1 crystal \(B(\Lambda)\) of the quantum affine algebras \(U_q(A_n^{(1)})\), and construct explicit crystal isomorphisms among them. The three realizations are polyhedral realizations (see [\textit{T. Nakashima} and \textit{A. Zelevinsky}, Adv. Math. 131, No. 1, 253--278 (1997; Zbl 0897.17014); \textit{T. Nakashima}, J. Algebra 219, No. 2, 571--597 (1999; Zbl 0967.17011)]), Nakajima monomial realizations (see [\textit{H. Nakajima}, Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop. Singapore: World Scientific. 196--219 (2001; Zbl 1011.17013); \textit{M. Kashiwara}, Contemp. Math. 325, 133--139 (2003; Zbl 1066.17006)]), and Young wall realizations (see [\textit{S.-J. Kang}, Proc. Lond. Math. Soc., III. Ser. 86, No. 1, 29--69 (2003; Zbl 1030.17013)]). They first give an explicit description of the connected component of the Nakajima monomial realization with special parameters, and show that this is isomorphic to a certain polyhedral realization. Using this isomorphism, they finally show the connections to Young wall realizations.
scientific article; zbMATH DE number 5931645
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