Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality
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- by Jun Hu and Zhankui Xiao;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 823-848
- DOI: https://doi.org/10.1090/btran/84
- Published electronically: September 14, 2021
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Abstract:
In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if $A$ is a quasi-hereditary algebra with a simple preserving duality and $T$ is a faithful tilting $A$-module, then $A$ has the double centralizer property with respect to $T$. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module $T$ over $A$ for which $A=End_{End_A(T)}(T)$. As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra $S_K^{sy}(m,n)$ and the Brauer algebra $\mathfrak {B}_n(-2m)$ on the space of dual partially harmonic tensors under certain condition.References
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Bibliographic Information
- Jun Hu
- Affiliation: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
- MR Author ID: 635795
- Email: junhu404@bit.edu.cn
- Zhankui Xiao
- Affiliation: School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, People’s Republic of China
- Email: zhkxiao@hqu.edu.cn
- Received by editor(s): November 29, 2020
- Received by editor(s) in revised form: March 22, 2021
- Published electronically: September 14, 2021
- Additional Notes: Zhankui Xiao is the corresponding author
The first author was supported by the National Natural Science Foundation of China. The second author was supported by the NSF of Fujian Province (Grant No. 2018J01002) and the National NSF of China (Grant No. 11871107). - © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 823-848
- MSC (2020): Primary 16D90, 20G05, 20G43, 05E10
- DOI: https://doi.org/10.1090/btran/84
- MathSciNet review: 4312325