AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Non-kissing complexes and tau-tilting for gentle algebras
About this Title
Yann Palu, Vincent Pilaud and Pierre-Guy Plamondon
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 274, Number 1343
ISBNs: 978-1-4704-5004-5 (print); 978-1-4704-6912-2 (online)
DOI: https://doi.org/10.1090/memo/1343
Published electronically: December 3, 2021
Keywords: Representations of gentle algebras,
support $\tau$-tilting theory,
Tamari lattice,
associahedron
Table of Contents
Chapters
- Acknowledgements
- Introduction
- 1. String modules
- 2. The non-kissing complex
- 3. The non-kissing lattice
- 4. The non-kissing associahedron
Abstract
We interpret the support $\tau$-tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its $\mathbf {g}$-vector fan and prove that it is the normal fan of a non-kissing associahedron.- Takuma Aihara and Osamu Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668. MR 2927802, DOI 10.1112/jlms/jdr055
- Takahide Adachi, Osamu Iyama, and Idun Reiten, $\tau$-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452. MR 3187626, DOI 10.1112/S0010437X13007422
- Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
- Yuliy Baryshnikov, On Stokes sets, New developments in singularity theory (Cambridge, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 21, Kluwer Acad. Publ., Dordrecht, 2001, pp. 65–86. MR 1849304
- Emily Barnard, The canonical join complex, Electron. J. Combin. 26 (2019), no. 1, Paper No. 1.24, 25. MR 3919619
- Thomas Brüstle, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, and Emine Yıldırım, On the combinatorics of gentle algebras, Canad. J. Math. 72 (2020), no. 6, 1551–1580. MR 4176701, DOI 10.4153/s0008414x19000397
- Anders Björner, Paul H. Edelman, and Günter M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), no. 3, 263–288. MR 1036875, DOI 10.1007/BF02187790
- M. C. R. Butler and Claus Michael Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179. MR 876976, DOI 10.1080/00927878708823416
- Thomas Brüstle and Dong Yang, Ordered exchange graphs, Advances in representation theory of algebras, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2013, pp. 135–193. MR 3220536
- Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), no. 4, 537–566. Dedicated to Robert V. Moody. MR 1941227, DOI 10.4153/CMB-2002-054-1
- F. Chapoton, Stokes posets and serpent nests, Discrete Math. Theor. Comput. Sci. 18 (2016), no. 3, Paper No. 18, 30. MR 3601367
- İ. Çanakçı, D. Pauksztello, and S. Schroll. On extensions for gentle algebras. In preparation, 2017.
- Ilke Canakci and Sibylle Schroll, Extensions in Jacobian algebras and cluster categories of marked surfaces, Adv. Math. 313 (2017), 1–49. With an appendix by Claire Amiot. MR 3649219, DOI 10.1016/j.aim.2017.03.016
- P. W. Donovan and M.-R. Freislich, The indecomposable modular representations of certain groups with dihedral Sylow subgroup, Math. Ann. 238 (1978), no. 3, 207–216. MR 514428, DOI 10.1007/BF01420248
- Aram Dermenjian, Christophe Hohlweg, and Vincent Pilaud, The facial weak order and its lattice quotients, Trans. Amer. Math. Soc. 370 (2018), no. 2, 1469–1507. MR 3729508, DOI 10.1090/tran/7307
- Laurent Demonet, Osamu Iyama, and Gustavo Jasso, $\tau$-tilting finite algebras, bricks, and $g$-vectors, Int. Math. Res. Not. IMRN 3 (2019), 852–892. MR 3910476, DOI 10.1093/imrn/rnx135
- L. Demonet, O. Iyama, N. Reading, I. Reiten, and H. Thomas. Lattice theory of torsion classes. Preprint, arXiv:1711.01785, 2017.
- Raika Dehy and Bernhard Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, Int. Math. Res. Not. IMRN 11 (2008), Art. ID rnn029, 17. MR 2428855, DOI 10.1093/imrn/rnn029
- Jesús A. De Loera, Jörg Rambau, and Francisco Santos, Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag, Berlin, 2010. Structures for algorithms and applications. MR 2743368, DOI 10.1007/978-3-642-12971-1
- Florian Eisele, Geoffrey Janssens, and Theo Raedschelders, A reduction theorem for $\tau$-rigid modules, Math. Z. 290 (2018), no. 3-4, 1377–1413. MR 3856858, DOI 10.1007/s00209-018-2067-4
- Ralph Freese, Jaroslav Ježek, and J. B. Nation, Free lattices, Mathematical Surveys and Monographs, vol. 42, American Mathematical Society, Providence, RI, 1995. MR 1319815, DOI 10.1090/surv/042
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
- I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1994 edition. MR 2394437
- A. Garver and T. McConville, Enumerative properties of Grid-Associahedra, Preprint, arXiv:1705.04901, 2017.
- Alexander Garver and Thomas McConville, Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions, J. Combin. Theory Ser. A 158 (2018), 126–175. MR 3800125, DOI 10.1016/j.jcta.2018.03.014
- I. M. Gel′fand and V. A. Ponomarev, Indecomposable representations of the Lorentz group, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3–60 (Russian). MR 0229751
- Christophe Hohlweg and Carsten E. M. C. Lange, Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), no. 4, 517–543. MR 2321739, DOI 10.1007/s00454-007-1319-6
- Christophe Hohlweg, Carsten E. M. C. Lange, and Hugh Thomas, Permutahedra and generalized associahedra, Adv. Math. 226 (2011), no. 1, 608–640. MR 2735770, DOI 10.1016/j.aim.2010.07.005
- Christophe Hohlweg, Vincent Pilaud, and Salvatore Stella, Polytopal realizations of finite type $\mathbf {g}$-vector fans, Adv. Math. 328 (2018), 713–749. MR 3771140, DOI 10.1016/j.aim.2018.01.019
- Jean-Louis Loday, Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), no. 3, 267–278. MR 2108555, DOI 10.1007/s00013-004-1026-y
- Carsten Lange and Vincent Pilaud, Associahedra via spines, Combinatorica 38 (2018), no. 2, 443–486. MR 3800847, DOI 10.1007/s00493-015-3248-y
- Thomas McConville, Lattice structure of grid-Tamari orders, J. Combin. Theory Ser. A 148 (2017), 27–56. MR 3603315, DOI 10.1016/j.jcta.2016.12.001
- Folkert Müller-Hoissen, Jean Marcel Pallo, and Jim Stasheff (eds.), Associahedra, Tamari lattices and related structures, Progress in Mathematics, vol. 299, Birkhäuser/Springer, Basel, 2012. Tamari memorial Festschrift. MR 3235205, DOI 10.1007/978-3-0348-0405-9
- Thibault Manneville and Vincent Pilaud, Geometric realizations of the accordion complex of a dissection, Discrete Comput. Geom. 61 (2019), no. 3, 507–540. MR 3918546, DOI 10.1007/s00454-018-0004-2
- T. Kyle Petersen, Pavlo Pylyavskyy, and David E. Speyer, A non-crossing standard monomial theory, J. Algebra 324 (2010), no. 5, 951–969. MR 2659207, DOI 10.1016/j.jalgebra.2010.05.001
- Vincent Pilaud, Pierre-Guy Plamondon, and Salvatore Stella, A $\tau$-tilting approach to dissections of polygons, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 045, 8. MR 3801898, DOI 10.3842/SIGMA.2018.045
- Vincent Pilaud and Francisco Santos, The brick polytope of a sorting network, European J. Combin. 33 (2012), no. 4, 632–662. MR 2864447, DOI 10.1016/j.ejc.2011.12.003
- Nathan Reading, Lattice and order properties of the poset of regions in a hyperplane arrangement, Algebra Universalis 50 (2003), no. 2, 179–205. MR 2037526, DOI 10.1007/s00012-003-1834-0
- Nathan Reading, Lattice congruences of the weak order, Order 21 (2004), no. 4, 315–344 (2005). MR 2209128, DOI 10.1007/s11083-005-4803-8
- Nathan Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110 (2005), no. 2, 237–273. MR 2142177, DOI 10.1016/j.jcta.2004.11.001
- Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313–353. MR 2258260, DOI 10.1016/j.aim.2005.07.010
- Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407–447. MR 2486939, DOI 10.4171/JEMS/155
- Ralf Schiffler, Quiver representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2014. MR 3308668, DOI 10.1007/978-3-319-09204-1
- Francisco Santos, Christian Stump, and Volkmar Welker, Noncrossing sets and a Grassmann associahedron, Forum Math. Sigma 5 (2017), Paper No. e5, 49. MR 3610869, DOI 10.1017/fms.2017.1
- James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
- Salvatore Stella, Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), no. 1, 121–158. MR 3070123, DOI 10.1007/s10801-012-0396-7
- Dov Tamari, Monoïdes préordonnés et chaînes de Malcev, Université de Paris, Paris, 1951 (French). Thèse. MR 0051833
- Burkhard Wald and Josef Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985), no. 2, 480–500. MR 801283, DOI 10.1016/0021-8693(85)90119-X
- Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1