Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux


Authors: Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara and Myungho Kim
Journal: Trans. Amer. Math. Soc. 366 (2014), 457-489
MSC (2010): Primary 17B37, 81R50
DOI: https://doi.org/10.1090/S0002-9947-2013-05866-7
Published electronically: September 19, 2013
MathSciNet review: 3118402
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give an explicit combinatorial realization of the crystal $ B(\lambda )$ for an irreducible highest weight $ U_q(\mathfrak{q}(n))$-module $ V(\lambda )$ in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms for decomposing the tensor product of $ \mathfrak{q}(n)$-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.


References [Enhancements On Off] (What's this?)

  • [1] Georgia Benkart, Seok-Jin Kang, and Masaki Kashiwara, Crystal bases for the quantum superalgebra $ U_q({\mathfrak{g}}{\mathfrak{l}}(m,n))$, J. Amer. Math. Soc. 13 (2000), no. 2, 295-331. MR 1694051 (2000m:17015), https://doi.org/10.1090/S0894-0347-00-00321-0
  • [2] Sergey Fomin, Schur operators and Knuth correspondences, J. Combin. Theory Ser. A 72 (1995), no. 2, 277-292. MR 1357774 (96k:05203), https://doi.org/10.1016/0097-3165(95)90065-9
  • [3] Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim, Quantum queer superalgebra and crystal bases, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 177-182. MR 2752232 (2011m:17037), https://doi.org/10.3792/pjaa.86.177
  • [4] D. Grantcharov, J. H. Jung, S.-J. Kang, M. Kashiwara, M. Kim,
    Crystal bases for the quantum queer superalgebra,
    to appear in J. Eur. Math. Soc. (JEMS).
  • [5] Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, and Myungho Kim, Highest weight modules over quantum queer superalgebra $ U_q(\mathfrak{q}(n))$, Comm. Math. Phys. 296 (2010), no. 3, 827-860. MR 2628823 (2011k:17026), https://doi.org/10.1007/s00220-009-0962-6
  • [6] Mark D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux, J. Combin. Theory Ser. A 50 (1989), no. 2, 196-225. MR 989194 (90j:05014), https://doi.org/10.1016/0097-3165(89)90015-0
  • [7] 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)
  • [8] Seok-Jin Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. London Math. Soc. (3) 86 (2003), no. 1, 29-69. MR 1971463 (2004c:17028), https://doi.org/10.1112/S0024611502013734
  • [9] Seok-Jin Kang and Jae-Hoon Kwon, Tensor product of crystal bases for $ U_q(\mathfrak{gl}(m,n))$-modules, Comm. Math. Phys. 224 (2001), no. 3, 705-732. MR 1871906 (2002k:17031), https://doi.org/10.1007/PL00005591
  • [10] Seok-Jin Kang and Jae-Hoon Kwon, Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm, J. Korean Math. Soc. 45 (2008), no. 4, 1135-1202. MR 2422732 (2009f:17022), https://doi.org/10.4134/JKMS.2008.45.4.1135
  • [11] Masaki Kashiwara, Crystalizing the $ q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249-260. MR 1090425 (92b:17018)
  • [12] 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), https://doi.org/10.1215/S0012-7094-91-06321-0
  • [13] Masaki Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839-858. MR 1240605 (95b:17019), https://doi.org/10.1215/S0012-7094-93-07131-1
  • [14] 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), https://doi.org/10.1006/jabr.1994.1114
  • [15] Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of $ U_q(\mathfrak{s}\mathfrak{l}(n))$, Comm. Math. Phys. 134 (1990), no. 1, 79-88. MR 1079801 (91j:17021)
  • [16] Toshiki Nakashima, Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras, Comm. Math. Phys. 154 (1993), no. 2, 215-243. MR 1224078 (94f:17015)
  • [17] G. I. Olshanski, Quantized universal enveloping superalgebra of type $ Q$ and a super-extension of the Hecke algebra, Lett. Math. Phys. 24 (1992), no. 2, 93-102. MR 1163061 (93i:17004), https://doi.org/10.1007/BF00402673
  • [18] Bruce E. Sagan, Shifted tableaux, Schur $ Q$-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), no. 1, 62-103. MR 883894 (88f:05011), https://doi.org/10.1016/0097-3165(87)90047-1
  • [19] A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras $ {\rm Gl}(n,\,m)$ and $ Q(n)$, Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422-430 (Russian). MR 735715 (85h:17010)
  • [20] Luis Serrano, The shifted plactic monoid, Math. Z. 266 (2010), no. 2, 363-392. MR 2678632 (2011i:05268), https://doi.org/10.1007/s00209-009-0573-0
  • [21] John R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87-134. MR 991411 (90k:20026), https://doi.org/10.1016/0001-8708(89)90005-4
  • [22] Dale Raymond Worley, A theory of shifted Young tableaux, ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)-Massachusetts Institute of Technology. MR 2941073

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B37, 81R50

Retrieve articles in all journals with MSC (2010): 17B37, 81R50


Additional Information

Dimitar Grantcharov
Affiliation: Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76021
Email: grandim@uta.edu

Ji Hye Jung
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: jhjung@math.snu.ac.kr

Seok-Jin Kang
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea
Email: sjkang@math.snu.ac.kr

Masaki Kashiwara
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan – and – Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: masaki@kurims.kyoto-u.ac.jp

Myungho Kim
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Email: mkim@math.snu.ac.kr, mhkim@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9947-2013-05866-7
Keywords: Quantum queer superalgebras, crystal bases, odd Kashiwara operators, semistandard decomposition tableaux, shifted Littlewood-Richardson coefficients
Received by editor(s): November 12, 2011
Received by editor(s) in revised form: May 1, 2012
Published electronically: September 19, 2013
Additional Notes: The first author was partially supported by NSA grant H98230-10-1-0207 and by the Research Institute for Mathematical Sciences, Kyoto University
The second author was partially supported by BK21 Mathematical Sciences Division and by NRF Grant # 2010-0010753
The third author was partially supported by KRF Grant # 2007-341-C00001, NRF Grant # 2010-0010753 and NRF Grant # 2010-0019516
The fourth author was partially supported by Grant-in-Aid for Scientific Research (B) 23340005, Japan Society for the Promotion of Science
The fifth author was partially supported by KRF Grant # 2007-341-C00001 and by NRF Grant # 2010-0019516
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society