Abstract
Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals \(\mathcal{B}(\lambda)\) for quantum affine algebras of type \(A_{\,n}^{(1)}\), \(B_n^{(1)}\), \(C_n^{(1)}\), \(A_{\,2n-1}^{(2)}\), \(A_{\,2n}^{(2)}\), and \(D_{n+1}^{(2)}\). The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.
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Hong, J., Kang, S.-J.: Crystal graphs for basic representations of the quantum affine algebra \({U}\sb q({C}\sp {(1)}\sb 2)\). Representations and Quantizations (Shanghai, 1998). China High. Educ., Beijing, pp. 213–227 (2000)
Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Grad. Stud. in Math. 42, Amer. Math. Soc. Providence, Rhode Island (2002)
Hong, J., Kang, S.-J., Lee, H.: Young wall realization of crystal graphs for \(U_q(C_n^{(1)})\). Comm. Math. Phys. 244(1), 111–131 (2004)
Jantzen, J.C.: Lectures on Quantum Groups. Grad. Stud. in Math. 6, Amer. Math. Soc., Providence, Rhode Island (1996)
Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of \({U}_q(\widehat{\mathfrak{sl}}(n))\) at \(q=0\). Comm. Math. Phys. 136(3), 543–566 (1991)
Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. London Math. Soc. 86(3), 29–69 (2003)
Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of Verma modules for quantum affine Lie algebras. Compositio Math. 92(3), 299–325 (1994)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Infinite analysis, Part A, B (Kyoto, 1991). World Scientific, River Edge, New Jersey, pp. 449–484 (1992)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68(3), 499–607 (1992)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Vertex models and crystals. C. R. Acad. Sci. Paris Sér. I Math. 315(4), 375–380 (1992)
Kashiwara, M.: On crystal bases of the \(q\)-analogue of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras. J. Algebra 165(2), 295–345 (1994)
Kuniba, A., Misra, K.C., Okado, M., Takagi, T., Uchiyama, J.: Crystals for Demazure modules of classical affine Lie algebras. J. Algebra 208(1), 185–215 (1998)
Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3(2), 447–498 (1990)
Misra, K., Miwa, T.: Crystal base for the basic representation of \({U}_q(\widehat{\mathfrak{sl}}(n))\). Comm. Math. Phys. 134(1), 79–88 (1990)
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Kang, SJ., Lee, H. Higher Level Affine Crystals and Young Walls. Algebr Represent Theor 9, 593–632 (2006). https://doi.org/10.1007/s10468-006-9013-6
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DOI: https://doi.org/10.1007/s10468-006-9013-6