Existence of Kirillov–Reshetikhin crystals for nonexceptional types

Okado, Masato; Schilling, Anne

doi:10.1090/S1088-4165-08-00329-4

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ISSN 1088-4165

Existence of Kirillov-Reshetikhin crystals for nonexceptional types

Authors: Masato Okado and Anne Schilling
Journal: Represent. Theory 12 (2008), 186-207
MSC (2000): Primary 17B37, 81R50; Secondary 05E15, 81R10
Posted: April 14, 2008
Erratum: Represent. Theory 12 (2008), 499--500
MathSciNet review: 2403558
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Abstract: Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals $B^{r,s}$ is established for all nonexceptional affine types. We also prove that the crystals $B^{r,s}$ of type $B_n^{(1)}$ , $D_n^{(1)}$ , and $A_{2n-1}^{(2)}$ are isomorphic to recently constructed combinatorial crystals for not a spin node.

1. Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335–402. MR 2066942 (2005e:17020)
2. Georgia Benkart, Igor Frenkel, Seok-Jin Kang, and Hyeonmi Lee, Level 1 perfect crystals and path realizations of basic representations at 𝑞=0, Int. Math. Res. Not. , posted on (2006), Art. ID 10312, 28. MR 2272099 (2007i:17030), http://dx.doi.org/10.1155/IMRN/2006/10312
3. Vyjayanthi Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, Internat. Math. Res. Notices 12 (2001), 629–654. MR 1836791 (2002i:17019), http://dx.doi.org/10.1155/S1073792801000332
4. Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59–78. MR 1357195 (96j:17009)
5. Vyjayanthi Chari and Andrew Pressley, Twisted quantum affine algebras, Comm. Math. Phys. 196 (1998), no. 2, 461–476. MR 1645027 (99f:16041), http://dx.doi.org/10.1007/s002200050431
6. P. Di Francesco, R. Kedem, Proof of the combinatorial Kirillov-Reshetikhin conjecture, preprint arXiv:0710.4415.
7. Goro Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi, and Zengo Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, Prog. Math. Phys., vol. 23, Birkhäuser Boston, Boston, MA, 2002, pp. 205–272. MR 1903978 (2003e:17020)
8. G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on fermionic formula, (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 243–291. MR 1745263 (2001m:81129), http://dx.doi.org/10.1090/conm/248/03826
9. David Hernandez, The Kirillov-Reshetikhin conjecture and solutions of 𝑇-systems, J. Reine Angew. Math. 596 (2006), 63–87. MR 2254805 (2007j:17020), http://dx.doi.org/10.1515/CRELLE.2006.052
10. D. Hernandez, Kirillov-Reshetikhin conjecture: The general case, preprint arXiv:0704.2838.
11. Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971 (2002m:17012)
12. Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
13. Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449–484. MR 1187560 (94a:17008)
14. Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499–607. MR 1194953 (94j:17013), http://dx.doi.org/10.1215/S0012-7094-92-06821-9
15. M. Kashiwara, On crystal bases of the 𝑄-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118 (93b:17045), http://dx.doi.org/10.1215/S0012-7094-91-06321-0
16. Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212 (95c:17024), http://dx.doi.org/10.1215/S0012-7094-94-07317-1
17. Masaki Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175. MR 1890649 (2002m:17013), http://dx.doi.org/10.1215/S0012-9074-02-11214-9
18. Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the 𝑞-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277 (95c:17025), http://dx.doi.org/10.1006/jabr.1994.1114
19. Yoshiyuki Koga, Level one perfect crystals for 𝐵⁽¹⁾_{𝑛},𝐶⁽¹⁾_{𝑛}, and 𝐷⁽¹⁾_{𝑛}, J. Algebra 217 (1999), no. 1, 312–334. MR 1700489 (2000h:17011), http://dx.doi.org/10.1006/jabr.1998.7806
20. Atsuo Kuniba and Tomoki Nakanishi, The Bethe equation at 𝑞=0, the Möbius inversion formula, and weight multiplicities. II. The 𝑋_{𝑛} case, J. Algebra 251 (2002), no. 2, 577–618. MR 1917385 (2003m:17013), http://dx.doi.org/10.1006/jabr.2001.8774
21. Atsuo Kuniba, Tomoki Nakanishi, and Zengo Tsuboi, The canonical solutions of the 𝑄-systems and the Kirillov-Reshetikhin conjecture, Comm. Math. Phys. 227 (2002), no. 1, 155–190. MR 1903843 (2003e:81085), http://dx.doi.org/10.1007/s002200200631
22. George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
23. Hiraku Nakajima, Extremal weight modules of quantum affine algebras, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 343–369. MR 2074599 (2005g:17036)
24. Hiraku Nakajima, 𝑡-analogs of 𝑞-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259–274 (electronic). MR 1993360 (2004e:17013), http://dx.doi.org/10.1090/S1088-4165-03-00164-X
25. H. Nakajima, -analogues of -characters of quantum affine algebras of type , preprint arXiv:math.QA/0606637.
26. Satoshi Naito and Daisuke Sagaki, Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras, Comm. Math. Phys. 263 (2006), no. 3, 749–787. MR 2211823 (2007a:17023), http://dx.doi.org/10.1007/s00220-005-1515-2
27. M. Okado, Existence of Crystal Bases for Kirillov-Reshetikhin Modules of Type , Publ. RIMS 43 (2007), 977-1004.
28. Masato Okado, Anne Schilling, and Mark Shimozono, Virtual crystals and fermionic formulas of type 𝐷⁽²⁾_{𝑛+1},𝐴⁽²⁾_{2𝑛}, and 𝐶⁽¹⁾_{𝑛}, Represent. Theory 7 (2003), 101–163 (electronic). MR 1973369 (2004f:17023), http://dx.doi.org/10.1090/S1088-4165-03-00155-9
29. Masato Okado, Anne Schilling, and Mark Shimozono, Virtual crystals and Kleber’s algorithm, Comm. Math. Phys. 238 (2003), no. 1-2, 187–209. MR 1989674 (2004c:17034)
30. A. Schilling, The combinatorial structure of Kirillov-Reshetikhin crystals of type $D_n^{(1)}$ , $B_n^{(1)}$ , $A_{2n-1}^{(2)}$ , J. Algebra, 319 (2008), 2938-2962.
31. Mark Shimozono, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin. 15 (2002), no. 2, 151–187. MR 1887233 (2002m:17005), http://dx.doi.org/10.1023/A:1013894920862
32. P. Sternberg, Applications of crystal bases to current problems in representation theory, Ph.D. thesis, UC Davis 2006 (available at arXiv:math.QA/0610704).
33. Michela Varagnolo and Eric Vasserot, Canonical bases and quiver varieties, Represent. Theory 7 (2003), 227–258 (electronic). MR 1990661 (2004i:17012), http://dx.doi.org/10.1090/S1088-4165-03-00154-7
34. Shigenori Yamane, Perfect crystals of 𝑈_{𝑞}(𝐺⁽¹⁾₂), J. Algebra 210 (1998), no. 2, 440–486. MR 1662347 (2000f:17024), http://dx.doi.org/10.1006/jabr.1998.7597