Finite-dimensional crystals B 2,s for quantum affine algebras of type D (1) n
Anne Schilling
and Philip Sternberg
University of California Department of Mathematics One Shields Avenue Davis CA 95616-8633 U.S.A. One Shields Avenue Davis CA 95616-8633 U.S.A
DOI: 10.1007/s10801-006-8347-9
Abstract
The Kirillov-Reshetikhin modules W r,s are finite-dimensional representations of quantum affine algebras U' q labeled by a Dynkin node r of the affine Kac-Moody algebra \mathfrak g \mathfrak{g} and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B 2,s corresponding to W 2,s for the algebra of type D (1) n.
Pages: 317–354
Keywords: keywords quantum affine algebras; crystal bases; kirillov-reshetikhin crystals
Full Text: PDF
References
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29. A. Schilling, A bijection between type D(1) n crystals and rigged configurations, J. Algebra 285(1) (2005), 292-334.
30. A. Schilling and M. Shimozono, X = M for symmetric powers, J. Algebra 295 (2006) 526-610.
2. V. Chari and A. Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994), 59-78, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.
3. V. Chari and A. Pressley, Twisted quantum affine algebras, Comm. Math. Phys. 196 (1998), 461-476.
4. V. G. Drinfeld, Hopf algebra and the Yang-Baxter equation, Soviet. Math. Dokl. 32 (1985), 254-258.
5. G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Z. Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, 205-272, Prog. Math. Phys., 23, Birkh\ddot auser Boston, Boston, MA, 2002.
6. G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), 243-291, Contemp. Math., 248, Amer. Math. Soc., Providence, RI,
1999. Springer J Algebr Comb (2006) 23: 317-354
7. J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics,
42. American Mathematical Society, Providence, RI, 2002. xviii+307 pp. ISBN: 0-8218-2874-6.
8. M. Jimbo, A q-difference analogue of U (G) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63-69.
9. V.G. Kac, Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp. ISBN: 0-521-37215-1.
10. S.-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68(3) (1992), 499-607.
11. S.-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991), 449-484, Adv. Ser. Math. Phys., 16, World Sci. Publishing, River Edge, NJ, 1992.
12. M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133(2) (1990), 249-260.
13. M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63(2) (1991), 465-516.
14. M. Kashiwara, On crystal bases, in: Representations of groups (Banff, AB, 1994), 155-197, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.
15. M. Kashiwara, On level-zero representation of quantized affine algebras, Duke Math. J. 112(1) (2002), 117-195.
16. M. Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41(1) (2005), 223-250.
17. M. Kashiwara and T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165(2) (1994), 295-345.
18. Y. Koga, Level one perfect crystals for B(1) n , C (1) n , and D(1) n , J. Algebra 217(1) (1999), 312-334.
19. S. V. Kerov, A. N. Kirillov and N. Y. Reshetikhin, Combinatorics, the Bethe ansatz and representations of the symmetric group, J. Soviet Math. 41(2) (1988), 916-924.
20. A. N. Kirillov and N. Y. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41(2) (1988), 925-955.
21. A. N. Kirillov, A. Schilling and M. Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations, Selecta Mathematica (N.S.) 8 (2002), 67-135.
22. C. Lecouvey, Schensted-type correspondences and plactic monoid for types Bn and Dn, J. Algebraic Combin. 18(2) (2003), 99-133.
23. G. Lusztig, Quantum deformation of certain simple modules over enveloping algebras, Adv. Math. 70(2) (1988), 237-249.
24. H. Nakajima, t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, Physics and combinatorics, 2000 (Nagoya), 196-219, World Sci. Publishing, River Edge, NJ, 2001.
25. M. Okado, A. Schilling and M. Shimozono, A crystal to rigged configuration bijection for nonexceptional affine algebras, in N. Jing (ed.) Algebraic Combinatorics and Quantum Groups, World Scientific (2003), 85-124.
26. M. Okado, A. Schilling and M. Shimozono, Virtual crystals and fermionic formulas of type D(2) , A(2) n+1 2 n , and C(1) n , Represent. Theory 7 (2003), 101-163.
27. M. Okado, A. Schilling and M. Shimozono, Virtual crystals and Kleber's algorithm, Commun. Math. Phys. 238 (2003), 187-209.
28. M. Shimozono, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin. 15(2) (2002), 151-187.
29. A. Schilling, A bijection between type D(1) n crystals and rigged configurations, J. Algebra 285(1) (2005), 292-334.
30. A. Schilling and M. Shimozono, X = M for symmetric powers, J. Algebra 295 (2006) 526-610.