Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

Hu, Jun; Xiao, Zhankui

doi:10.1090/btran/84

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality
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by Jun Hu and Zhankui Xiao HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 823-848

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.

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