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Transactions of the American Mathematical Society
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
Published electronically: September 19, 2013
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-2013-05866-7
PII: S 0002-9947(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