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ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

Part of: Representation theory of rings and algebras Algebraic combinatorics Abelian categories

Published online by Cambridge University Press:  07 February 2019

ALEXANDER GARVER Open the ORCID record for ALEXANDER GARVER [Opens in a new window]  and
THOMAS MCCONVILLE
ALEXANDER GARVER*
Affiliation:
Université du Québec à Montréal, Montréal, Canada e-mail: [email protected]
THOMAS MCCONVILLE
Affiliation:
Mathematical Sciences Research Institute, Berkeley, CA, USA e-mail: [email protected]
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Abstract

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The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 05E10: Combinatorial aspects of representation theory 18E40: Torsion theories, radicals
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 147 - 182
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

References

Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras: Techniques of representation theory (London Mathematical Society Student Texts 65), vol. 1 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
Baur, K. and Coelho Simões, R., A geometric model for the module category of a gentle algebra. arXiv:1803.05802 (2018).Google Scholar
Bridgeland, T., Stability conditions on triangulated categories. Ann. of Math. (2) 166(2) (2007), 317–345.Google Scholar
Brüstle, T., Douville, G., Mousavand, K., Thomas, H. and Yıldrm, E., On the combinatorics of gentle algebras. arXiv:1707.07665 (2017).Google Scholar
Brüstle, T. and Yang, D., Ordered exchange graphs. Adv. Rep. Theor. Algeb. (ICRA Bielefeld 2012). (2013), 135–193.CrossRefGoogle Scholar
Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics. Adv. Math. 204(2) (2006), 572–618.CrossRefGoogle Scholar
Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (An case). Trans. Amer. Math. Soc. 358(3) (2006), 1347–1364.CrossRefGoogle Scholar
Canacki, I. and Schroll, S., Extensions in Jacobian algebras and cluster categories of marked surfaces. Adv. Math.313 (2017), 1–49.Google Scholar
Canakci, I., Pauksztello, D. and Schroll, S., Mapping cones in the bounded derived category of a gentle algebra. arXiv:1609.09688 (2016).Google Scholar
Canakci, I., Pauksztello, D. and Schroll, S., On extensions for gentle algebras. arXiv:1707.06934 (2017).Google Scholar
Chavez, A. N., c-vectors and dimension vectors for cluster-finite quivers. Bull. Lond. Math. Soc. (2013), 1590–1600.Google Scholar
Coelho Simões, R. and Parsons, M. J., Endomorphism algebras for a class of negative Calabi–Yau categories. J. Algeb.491 (2017), 32–57.CrossRefGoogle Scholar
Demonet, L., Algebras of partial triangulations. arXiv:1602.01592 (2016).Google Scholar
Demonet, L., Iyama, O., Reading, N., Reiten, I. and Thomas, H., Lattice theory of torsion classes. arXiv:1711.01785 (2017).Google Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations I: mutations. Selecta Math. 14(1) (2008), 59–119.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras I: foundations. J. Amer. Math. Soc. 15(2) (2002), 497–529.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras IV: coefficients. Comp. Math. 143(01) (2007), 112–164.CrossRefGoogle Scholar
Garver, A., Igusa, K., Matherne, J. P. and Ostroff, J., Combinatorics of exceptional sequences in type A. Electron. J. Combin. (in press).Google Scholar
Garver, A. and McConville, T., Lattice properties of oriented exchange graphs and torsion classes. Algeb. Rep. Theor. (in press), 1–36.Google Scholar
Garver, A. and McConville, T., Oriented flip graphs and noncrossing tree partitions. arXiv:1604.06009 (2016).Google Scholar
Garver, A. and McConville, T., Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions. J. Comb. Theor. Ser. A 158 (2018), 126–175.CrossRefGoogle Scholar
Grätzer, G. and Wehrung, F., Lattice theory: special topics and applications. vol. 2 (Springer, Switzerland, 2016).CrossRefGoogle Scholar
Happel, D., Reiten, I. and Smalø, S. O., Tilting in Abelian categories and quasitilted algebras, vol. 575 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
Ingalls, C. and Thomas, H., Noncrossing partitions and representations of quivers. Compos. Math. 145(06) (2009), 1533–1562.Google Scholar
Iyama, O., Reiten, I., Thomas, H. and Todorov, G., Lattice structure of torsion classes for path algebras. Bull. Lond. Math. Soc. 47(4) (2015), 639–650.Google Scholar
Kashiwara, M. and Schapira, P., Sheaves on manifolds, vol. 292 (Springer Science & Business Media, Berlin, Heidelberg, 2013).Google Scholar
Koenig, S. and Yang, D., Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras. Doc. Math.19 (2014), 403–438.Google Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. arXiv:0811.2435 (2008).Google Scholar
Marks, F. and Št’ovíček, J., Torsion classes, wide subcategories and localisations. Bull. Lond. Math. Soc.49(3) (2017), 405–416.CrossRefGoogle Scholar
Murphy, G. J.. Derived equivalence classification of m-cluster tilted algebras of type An. J. Algeb. 323(4) (2010), 920–965.CrossRefGoogle Scholar
Opper, S., Plamondon, P.-G. and Schroll, S., A geometric model for the derived category of gentle algebras, arXiv:1801.09659 (2018).Google Scholar
Palu, Y., Pilaud, V. and P.-G. Plamondon., Non-kissing complexes and tau-tilting for gentle algebras. Mem. Amer. Math. Soc. (in press).Google Scholar
Reading, N., Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Amer. Math. Soc. 359(12) (2007), 5931–5958.CrossRefGoogle Scholar
Rickard, J., Equivalences of derived categories for symmetric algebras. J. Algeb. 257(2) (2002), 460–481.CrossRefGoogle Scholar
Ringel, C. M., Representations of k-species and bimodules. J. Algeb. 41(2) (1976), 269–302.CrossRefGoogle Scholar
Speyer, D. and Thomas, H., Acyclic cluster algebras revisited, in Algebras, quivers and representations (vol. 8 of Abel Symp., Springer, Berlin, Heidelberg, 2013), 275–298.CrossRefGoogle Scholar
Wald, B. and Waschbusch, J., Tame biserial algebras. J. Algeb. 95(2) (1985), 480–500.CrossRefGoogle Scholar
Woolf, J., Stability conditions, torsion theories and tilting. J. Lond. Math. Soc. 82(3) (2010), 663–682.Google Scholar