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arXiv · 1710.10375

The $q$-Schur algebras and $q$-Schur dualities of finite type

Li Luo·Weiqiang Wang
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Abstract

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra and Hecke algebra associated to $W$. We then realize geometrically the $q$-Schur algebra and duality, and construct a canonical basis for the $q$-Schur algebra with positivity. With suitable choices of $X_{\texttt f}$ in classical types, we recover the $q$-Schur algebras in the literature. Our $q$-Schur algebras are closely related to the category $\mathcal O$, where the type $G_2$ is studied in detail.

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BibTeXRIS

Li Luo, Weiqiang Wang. 2017-10-28. The $q$-Schur algebras and $q$-Schur dualities of finite type. https://arxiv.org/abs/1710.10375

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