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The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

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Abstract

Canonical bases of the tensor powers of the natural \( U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) \)-module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the ℤ2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra O q (M m|n ) of a quantum (m,n) × (m,n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for \( U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) \) by applying a quantum analogue of the Borel-Weil construction.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    HeChun Zhang

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  1. HeChun Zhang
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Correspondence to HeChun Zhang.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10471070)

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Zhang, H. The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials. Sci. China Ser. A-Math. 52, 401–416 (2009). https://doi.org/10.1007/s11425-008-0150-8

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  • DOI: https://doi.org/10.1007/s11425-008-0150-8

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