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Canonical bases for the quantum supergroups \({\mathbf {U}}(\mathfrak {gl}_{m|n})\)

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Abstract

We give a combinatorial construction for the canonical bases of the \(\pm \)-parts of the quantum enveloping superalgebra \({\mathbf {U}}(\mathfrak {gl}_{m|n})\) and discuss their relationship with the Kazhdan–Lusztig bases for the quantum Schur superalgebras \({\varvec{{\mathcal {S}}}}(m|n,r)\) introduced in Du and Rui (J Pure Appl Algebra 215:2715–2737, 2011). We will also extend this relationship to the induced bases for simple polynomial representations of \({\mathbf {U}}(\mathfrak {gl}_{m|n})\).

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Notes

  1. We have corrected some typos given in [8, (6.2.1)].

  2. This order relation is denoted by \(\sqsubseteq \) in [1].

  3. We have corrected typos in line 3 from [7, Th. 8.4].

  4. Since \({\lambda }-\mu =({\lambda }_1-\mu _1){\varvec{e}}_1+\cdots +({\lambda }_n-\mu _n){\varvec{e}}_n=(\tilde{\lambda }_1-\tilde{\mu }_1)({\varvec{e}}_1-{\varvec{e}}_2)+\cdots +(\tilde{\lambda }_{n-1}-\tilde{\mu }_{n-1})({\varvec{e}}_{n-1}-{\varvec{e}}_n)\) (\(\tilde{a}_j=\sum _{i=1}^ja_i\)), this order is the usual dominance order \(\unrhd \) if \({\lambda },\mu \) are regarded as compositions.

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Acknowledgments

The authors would like to thank Weiqiang Wang for the reference [5]. The first author also thanks him for various discussions during his visit to Charlottesville in January 2014 and for his comments on the canonical property in the \(\mathfrak {gl}_{m|1}\) case. Thanks are also due to the referee for thoroughly reading the paper and for many helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Jie Du

  2. Department of Mathematics, East China Normal University, Shanghai, 200062, China

    Haixia Gu

  3. Huzhou University, Huzhou, 313000, Zhejiang, China

    Haixia Gu

Authors
  1. Jie Du
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  2. Haixia Gu
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Correspondence to Haixia Gu.

Additional information

The authors gratefully acknowledge support from ARC under Grant DP120101436, NSFC (No. 11131001) and ZJNSF (Nos. LQ12A010001, LZ14A010001). The work was completed while the second author was visiting UNSW.

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Du, J., Gu, H. Canonical bases for the quantum supergroups \({\mathbf {U}}(\mathfrak {gl}_{m|n})\) . Math. Z. 281, 631–660 (2015). https://doi.org/10.1007/s00209-015-1499-3

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