Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$
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- by Satoshi Naito and Daisuke Sagaki PDF
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Abstract:
Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of a (general) integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty }$, $C_{\infty }$, or $D_{\infty }$. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results in types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$ extend the corresponding results due to Kwon in types $A_{+\infty }$ and $A_{\infty }$. Our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey in types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$, where the extremal weights are of level zero.References
- Tatsuya Akasaka and Masaki Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, 839–867. MR 1607008, DOI 10.2977/prims/1195145020
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- 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, DOI 10.1090/gsm/042
- Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966, DOI 10.1007/978-3-642-78400-2
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Masaki Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858. MR 1240605, DOI 10.1215/S0012-7094-93-07131-1
- Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212, DOI 10.1215/S0012-7094-94-07317-1
- Masaki Kashiwara, Similarity of crystal bases, Lie algebras and their representations (Seoul, 1995) Contemp. Math., vol. 194, Amer. Math. Soc., Providence, RI, 1996, pp. 177–186. MR 1395599, DOI 10.1090/conm/194/02393
- Masaki Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175. MR 1890649, DOI 10.1215/S0012-9074-02-11214-9
- Kazuhiko Koike, Representations of spinor groups and the difference characters of $\textrm {SO}(2n)$, Adv. Math. 128 (1997), no. 1, 40–81. MR 1451419, DOI 10.1006/aima.1997.1625
- Kazuhiko Koike, On representation of the classical groups [ MR1418849 (97m:20053)], Selected papers on harmonic analysis, groups, and invariants, Amer. Math. Soc. Transl. Ser. 2, vol. 183, Amer. Math. Soc., Providence, RI, 1998, pp. 79–100. MR 1615138, DOI 10.1090/trans2/183/04
- Jae-Hoon Kwon, Demazure crystals of generalized Verma modules and a flagged RSK correspondence, J. Algebra 322 (2009), no. 6, 2150–2179. MR 2542836, DOI 10.1016/j.jalgebra.2009.04.034
- Jae-Hoon Kwon, Differential operators and crystals of extremal weight modules, Adv. Math. 222 (2009), no. 4, 1339–1369. MR 2554938, DOI 10.1016/j.aim.2009.06.012
- Jae-Hoon Kwon, Crystal duality and Littlewood-Richardson rule of extremal weight crystals, J. Algebra 336 (2011), 99–138. MR 2802533, DOI 10.1016/j.jalgebra.2011.04.010
- Jae-Hoon Kwon, Crystal bases of modified quantized enveloping algebras and a double RSK correspondence, J. Combin. Theory Ser. A 118 (2011), no. 7, 2131–2156. MR 2802192, DOI 10.1016/j.jcta.2011.04.006
- Cédric Lecouvey, Crystal bases and combinatorics of infinite rank quantum groups, Trans. Amer. Math. Soc. 361 (2009), no. 1, 297–329. MR 2439408, DOI 10.1090/S0002-9947-08-04480-2
- Peter Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3, 329–346. MR 1253196, DOI 10.1007/BF01231564
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- S. Natio and D. Sagaki, Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{\infty }$, $C_{\infty }$, and $D_{\infty }$, preprint 2010, arXiv:1003.2485.
Additional Information
- Satoshi Naito
- Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
- Email: naito@math.titech.ac.jp
- Daisuke Sagaki
- Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
- MR Author ID: 680572
- Email: sagaki@math.tsukuba.ac.jp
- Received by editor(s): May 11, 2010
- Received by editor(s) in revised form: December 29, 2010, and April 1, 2011
- Published electronically: May 29, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6531-6564
- MSC (2010): Primary 17B37; Secondary 05E10, 05A19, 17B67
- DOI: https://doi.org/10.1090/S0002-9947-2012-05597-8
- MathSciNet review: 2958947