Abstract
Semistable subcategories were introduced in the context of Mumford’s GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding poset of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Coelho Simões and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Moreover, this bijection defines an isomorphism of the posets on these objects. Our work recovers that of Ingalls and Thomas in Dynkin type A.
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Notes
For a general finite dimensional \(\Bbbk \)-algebra \(\Lambda = \Bbbk Q/I\) where I is an admissible ideal, one can also equivalently describe modules over \(\Lambda \) as representations of Q compatible with I.
For simplicity, we have given the definition of M(w) only in the generality of tiling algebras.
Up to coloring-preserving isotopy relative to z(v, F) and z(u, G), there is a unique green (resp., red) admissible curve for [v, u].
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Acknowledgements
This project began at a Mitacs Globalink Research Internship at Université du Québec à Montréal. M. Garcia was supported by Mitacs Globalink and the project CONACyT-238754. A. Garver was supported by NSERC Grant RGPIN/05999-2014 and the Canada Research Chairs Program. The authors thank an anonymous referee for careful comments that helped to improve the manuscript. The authors also thank Yann Palu for important input on the proof of Theorem 1.1.
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Garcia, M., Garver, A. Semistable subcategories for tiling algebras. Beitr Algebra Geom 61, 47–71 (2020). https://doi.org/10.1007/s13366-019-00461-y
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DOI: https://doi.org/10.1007/s13366-019-00461-y