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Cells in affine q-Schur algebras

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Abstract

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.

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Acknowledgement

We thank Geordie Williamson for helpful discussions on the inner product for Schur algebroids. W. C. is partially supported by China Scholarship Council, Young Scholars Program of Shandong University and the NSF of China (grant No. 11601273). W. C. thanks University of Virginia for hospitality and support. L. L. is partially supported by the Science and Technology Commission of Shanghai Municipality (grant Nos. 21ZR1420000, 22DZ222901) and the NSF of China (grant No. 12371028), and Fundamental Research Funds for the Central Universities. W. W. is partially supported by the NSF grant DMS-1702254 and DMS-2001351.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

    Weideng Cui

  2. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

    Li Luo

  3. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Weiqiang Wang

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  1. Weideng Cui
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  2. Li Luo
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Correspondence to Li Luo.

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Cui, W., Luo, L. & Wang, W. Cells in affine q-Schur algebras. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2620-2

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  • DOI: https://doi.org/10.1007/s11856-024-2620-2

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