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Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type

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We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.

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Acknowledgments

First and foremost, the author would like to thank Osamu Iyama for his support and patient guidance. He would like to express his gratitude to Steffen Oppermann, who kindly explain results of his paper. He is grateful to Joseph Grant, Laurent Demonet and Dong Yang for answering questions and helpful comments. He thanks Kota Yamaura, Takahide Adachi and Gustavo Jasso for their help and stimulating discussions. He is very grateful to the anonymous referee for valuable comments, especially for suggesting the better terms and sentences.

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  1. Graduate School of Mathematics, Nagoya University, Frocho, Chikusaku, Nagoya, 464-8602, Japan

    Yuya Mizuno

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  1. Yuya Mizuno
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Correspondence to Yuya Mizuno.

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The author is supported by Grant-in-Aid for JSPS Fellowships No. 23.5593.

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Mizuno, Y. Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type. Math. Z. 277, 665–690 (2014). https://doi.org/10.1007/s00209-013-1271-5

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