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THE $q$-SCHUR ALGEBRAS AND $q$-SCHUR DUALITIES OF FINITE TYPE
Published online by Cambridge University Press: 19 February 2020
Abstract
We formulate a $q$-Schur algebra associated with an arbitrary $W$-invariant finite set $X_{\text{f}}$ 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 with $W$. 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 $X_{\text{f}}$ 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.
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 129 - 160
- Copyright
- © The Author(s) 2020. Published by Cambridge University Press
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