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PBW Bases and Marginally Large Tableaux in Types B and C

Part of: Algebraic combinatorics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  04 January 2019

Jackson A. Criswell ,
Ben Salisbury  and
Peter Tingley
Jackson A. Criswell
Affiliation:
Central Michigan University, Mount Pleasant, MI 48859, USA Email: crisw1ja@cmich.edusalis1bt@cmich.edu
Ben Salisbury
Affiliation:
Central Michigan University, Mount Pleasant, MI 48859, USA Email: crisw1ja@cmich.edusalis1bt@cmich.edu
Peter Tingley
Affiliation:
Loyola University Chicago, Chicago, IL 60660, USA Email: ptingley@luc.edu
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Abstract

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We explicitly describe the isomorphism between two combinatorial realizations of Kashiwara’s infinity crystal in types B and C. The first realization is in terms of marginally large tableaux and the other is in terms of Kostant partitions coming from PBW bases. We also discuss a stack notation for Kostant partitions which simplifies that realization.

Keywords

crystalKostant partitionLusztig datamarginally large tableau

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 37 - 54
Copyright
© Canadian Mathematical Society 2018 

Footnotes

B.S. was partially supported by Simons Foundation grant #429950. P.T. was partially supported by NSF grant DMS-1265555.

References

Berenstein, A. and Zelevinsky, A., Tensor product multiplicities, canonical bases and totally positive varieties . Invent. Math. 143(2001), no. 1, 77–128. https://doi.org/10.1007/s002220000102.Google Scholar
Bourbaki, N., Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.Google Scholar
Claxton, J. and Tingley, P., Young tableaux, multisegments, and PBW bases . Sém. Lothar. Combin. 73(2015), Article B73c.Google Scholar
Cliff, G., Crystal bases and Young tableaux . J. Algebra 202(1998), no. 1, 10–35. https://doi.org/10.1006/jabr.1997.7244.Google Scholar
Hong, J. and Lee, H., Young tableaux and crystal B(∞) for finite simple Lie algebras . J. Algebra 320(2008), no. 10, 3680–3693. https://doi.org/10.1016/j.jalgebra.2008.06.008.Google Scholar
Jacon, N. and Lecouvey, C., Kashiwara and Zelevinsky involutions in affine type A . Pacific J. Math. 243(2009), no. 2, 287–311. https://doi.org/10.2140/pjm.2009.243.287.Google Scholar
Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras . Duke Math. J. 63(1991), no. 2, 465–516. https://doi.org/10.1215/S0012-7094-91-06321-0.Google Scholar
Kwon, J.-H., Lusztig data of Kashiwara–Nakashima tableaux in types B and C. 2017. arxiv:1610.02640.Google Scholar
Leclerc, B., Thibon, J.-Y., and Vasserot, E., Zelevinsky’s involution at roots of unity . J. Reine Angew. Math. 513(1999), 33–51.Google Scholar
Lusztig, G., Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010.10.1007/978-0-8176-4717-9Google Scholar
Salisbury, B., Schultze, A., and Tingley, P., Combinatorial descriptions of the crystal structure on certain PBW bases . Transform. Groups, to appear. arxiv:1606.01978.Google Scholar
Salisbury, B., Schultze, A., and Tingley, P., PBW bases and marginally large tableaux in type D . J. Comb., to appear. arxiv:1606.02517.Google Scholar
Tingley, P., Elementary construction of Lusztig’s canonical basis . Groups, rings, group rings and Hopf algebras, Contemp. Math. 688(2017), 265–277.10.1090/conm/688/13839Google Scholar
Zelevinsky, A. V., Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n) . Ann. Sci. École Norm. Sup. (4) 13(1980), no. 2, 165–210. https://doi.org/10.24033/asens.1379.Google Scholar