Torsion pairs for quivers and the Weyl groups

Abstract

We give an interpretation of the map \(\pi ^c\) defined by Reading, which is a map from the elements of a Coxeter group to the c-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close relationship between c-sortable elements and torsion pairs, and give an explicit description of the cofinite torsion classes in the context of the Coxeter group. As a consequence, we give a proof of some conjectures proposed by Oppermann, Reiten, and the second author.

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Acknowledgements

The first author would like to thank Osamu Iyama and Takahide Adachi for their valuable comments and stimulating discussions. The second author would like to thank Steffen Oppermann and Idun Reiten for their collaboration on [19]. Both authors would also like to express their appreciation to the Mathematisches Forschungsinstitut Oberwolfach, where our collaboration began, and to the referee, whose suggestions substantially improved the paper.

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Correspondence to Hugh Thomas.

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Yuya Mizuno is supported by Grant-in-Aid for JSPS Research Fellow 17J00652. Hugh Thomas is supported by an NSERC Discovery Grant and the Canada Reseach Chairs program.

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Mizuno, Y., Thomas, H. Torsion pairs for quivers and the Weyl groups. Sel. Math. New Ser. 26, 46 (2020). https://doi.org/10.1007/s00029-020-00563-9

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Mathematics Subject Classification

  • 16G10
  • 16G20
  • 20F55