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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Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-tilting theory. Compos. Math. 150(3), 415–452 (2014)
Amiot, C., Iyama, O., Reiten, I., Todorov, G.: Preprojective algebras and \(c\)-sortable words. Proc. Lond. Math. Soc. 104(3), 513–539 (2012)
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006)
Baumann, P., Kamnitzer, J.: Preprojective algebras and MV polytopes. Represent. Theory 16, 152–188 (2012)
Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirković–Vilonen polytopes. Publ. Math. Inst. Hautes Etudes Sci. 120, 113–205 (2014)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 4–6 , Elements of Mathematics (Berlin). Springer, Berlin, (2002). Translated from the 1968 French original by Andrew Pressley
Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145, 1035–1079 (2009)
Geiss, C., Leclerc, B., Schröer, J.: Kac–Moody groups and cluster algebras. Adv. Math. 228(1), 329–433 (2011)
Hohlweg, C., Lange, C., Thomas, H.: Permutahedra and generalized associahedra. Adv. Math. 226, 608–640 (2011)
Humphreys, J.E.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge (1990)
Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533–1562 (2009)
Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Am. J. Math. 130(4), 1087–1149 (2008)
Iyama, O., Reiten, I.: 2-Auslander algebras associated with reduced words in Coxeter groups. Int. Math. Res. Not. IMRN 8, 1782–1803 (2011)
Kac, V.G.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)
Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J. 89(1), 9–36 (1997)
Leclerc, B.: Cluster structures on strata of flag varieties. Adv. Math. 300, 190–228 (2016)
Mizuno, Y.: Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type. Math. Z. 277(3), 665–690 (2014)
Oppermann, S., Reiten, I., Thomas, H.: Quotient closed subcategories of quiver representations. Compos. Math. 151(3), 568–602 (2015)
Reading, N.: Cambrian Lattices. Adv. Math. 205(2), 313–353 (2006)
Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359(12), 5931–5958 (2007)
Reading, N.: Sortable elements and Cambrian lattices. Algebra Univ. 56(3–4), 411–437 (2007)
Reading, N., Speyer, D.: Cambrian fans. J. Eur. Math. Soc. (JEMS) 11(2), 407–447 (2009)
Reading, N., Speyer, D.: Sortable elements in infinite Coxeter groups. Trans. Am. Math. Soc. 363(2), 699–761 (2011)
Sekiya, Y., Yamaura, K.: Tilting theoretical approach to moduli spaces over preprojective algebras. Algebras Represent. Theory 16(6), 1733–1786 (2013)
Smalø, S.O.: Torsion theory and tilting modules. Bull. Lond. Math. Soc. 16, 518–522 (1984)
Thomas, H.: Coxeter groups and quiver representations. Surveys in representation theory of algebras. Contemp. Math. 716, 173–186, Amer. Math. Soc., Providence, RI (2018)
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 . 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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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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