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Adjoint crystals and Young walls for \(U_q(\widehat{sl}_2)\). (English) Zbl 1202.17005

The authors develop the combinatorics of Young walls associated with higher level adjoint crystals for the quantum affine algebra \(U_q(\widehat{sl}_2)\). The irreducible highest weight crystal \(B(\lambda )\) of arbitrary level is realized as the affine crystal consisting of reduced Young walls on \(\lambda \). A Young wall realization of the crystal \(B(\infty )\) for \(U_q^-(\widehat{sl}_2)\) is also given.
Reviewer: Hu Jun (Beijing)

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E10 Combinatorial aspects of representation theory
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[1] Benkart, G.; Frenkel, I.; Kang, S.-J.; Lee, H., Level 1 perfect crystals and path realizations of basic representations at \(q = 0\), Int. Math. Res. Not., 2006, 1-28 (2006) · Zbl 1149.17016
[2] G. Fourier, M. Okado, A. Schilling, Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types, arXiv:0811.1604v1 [math.RT]; G. Fourier, M. Okado, A. Schilling, Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types, arXiv:0811.1604v1 [math.RT] · Zbl 1267.17012
[3] Hong, J.; Kang, S.-J., (Introduction to Quantum Groups and Crystal Bases. Introduction to Quantum Groups and Crystal Bases, Graduate Studies in Mathematics, vol. 42 (2002), American Mathematical Society) · Zbl 1134.17007
[4] Jimbo, M.; Misra, K. C.; Miwa, T.; Okado, M., Combinatorics of representations of \(U_q(\hat{s l}(n))\) at \(q = 0\), Comm. Math. Phys., 136, 543-566 (1991) · Zbl 0749.17015
[5] Kang, S.-J., Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. Lond. Math. Soc. (3), 86, 29-69 (2003) · Zbl 1030.17013
[6] S.-J. Kang, Perfect crystals and Young walls (in preparation); S.-J. Kang, Perfect crystals and Young walls (in preparation)
[7] Kang, S.-J.; Kashiwara, M.; Misra, K. C., Crystal bases of Verma modules for quantum affine Lie algebras, Compos. Math., 92, 299-325 (1994) · Zbl 0808.17007
[8] Kang, S.-J.; Kashiwara, M.; Misra, K. C.; Miwa, T.; Nakashima, T.; Nakayashiki, A., Affine crystals and vertex models, Int. J. Mod. Phys. A., Suppl. 1A, 449-484 (1992) · Zbl 0925.17005
[9] 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, 499-607 (1992) · Zbl 0774.17017
[10] Kang, S.-J.; Lee, H., Crystal bases for quantum affine algebras and Young walls, J. Algebra, 322, 1979-1999 (2009) · Zbl 1196.17014
[11] Kashiwara, M.; Miwa, T.; Peterson, J.-U. H.; Yung, C. M., Perfect crystals and \(q\)-deformed Fock space, Selecta Math., 2, 415-499 (1996) · Zbl 0959.17014
[12] M. Kim, S. Kim, On the connection between Young walls and Littelmann paths, preprint (KIAS-M09012: see http://www.kias.re.kr/en/programs/math_pub_list.jsp), August 2009; M. Kim, S. Kim, On the connection between Young walls and Littelmann paths, preprint (KIAS-M09012: see http://www.kias.re.kr/en/programs/math_pub_list.jsp), August 2009
[13] Littelmann, P., A Littlewood-Richardson rule for symemtrizable Kac-Moody algebras, Invent. Math., 116, 326-346 (1994) · Zbl 0805.17019
[14] Littelmann, P., Paths and root operators in representation theory, Ann. of Math., 3, 142, 499-525 (1995) · Zbl 0858.17023
[15] Schilling, A.; Sternberg, P., Finite-dimensional crystals \(B^{2, s}\) for quantum affine algebras of type \(D_n^{(1)}\), J. Algebraic Combin., 23, 317-354 (2006) · Zbl 1195.17014
[16] J.-Y. Yu, A new realization of crystal graphs for \(A_1^{( 1 )}\), M.S. Thesis, Seoul National University, 2007; J.-Y. Yu, A new realization of crystal graphs for \(A_1^{( 1 )}\), M.S. Thesis, Seoul National University, 2007
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