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arXiv · 2109.10537

Geometric Howe dualities of finite type

Li Luo·Zheming Xu
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Abstract

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that the quantum Howe duality derived from partial flag varieties coincides with the one constructed by quantum coordinate (co)algebras. Moreover, the explicit multiplicity-free decompositions for these Howe dualities are obtained.

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Li Luo, Zheming Xu. 2021-09-22. Geometric Howe dualities of finite type. https://arxiv.org/abs/2109.10537

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