Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 045, 8 pages      arXiv:1710.02119      https://doi.org/10.3842/SIGMA.2018.045

A $\tau$-Tilting Approach to Dissections of Polygons

Vincent Pilaud a, Pierre-Guy Plamondon b and Salvatore Stella c
a) CNRS & LIX, École Polytechnique, Palaiseau, France
b) Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, France
c) University of Haifa, Israel

Received February 26, 2018, in final form May 10, 2018; Published online May 14, 2018

Abstract
We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $\tau$-tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced subcomplexes of support $\tau$-tilting simplicial complexes of finite dimensional algebras.

Key words: dissections of polygons; accordion complexes; $\tau$-tilting theory; representations of finite dimensional algebras; $\mathbf{g}$-vectors.

pdf (370 kb)   tex (270 kb)

References

  1. Adachi T., Iyama O., Reiten I., $\tau$-tilting theory, Compos. Math. 150 (2014), 415-452, arXiv:1210.1036.
  2. Assem I., Simson D., Skowroński A., Elements of the representation theory of associative algebras, Vol. 1, Techniques of representation theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006.
  3. Buan A.B., Marsh R., Reineke M., Reiten I., Todorov G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572-618, math.RT/0402054.
  4. Butler M.C.R., Ringel C.M., Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), 145-179.
  5. Caldero P., Chapoton F., Schiffler R., Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math. Soc. 358 (2006), 1347-1364.
  6. Chapoton F., Stokes posets and serpent nests, Discrete Math. Theor. Comput. Sci. 18 (2016), 18, 30 pages, arXiv:1505.05990.
  7. David-Roesler L., Derived equivalence of surface algebras in genus 0 via graded equivalence, Algebr. Represent. Theory 17 (2014), 1-30, arXiv:11111.4657.
  8. David-Roesler L., Schiffler R., Algebras from surfaces without punctures, J. Algebra 350 (2012), 218-244, arXiv:1103.4357.
  9. Dehy R., Keller B., On the combinatorics of rigid objects in 2-Calabi-Yau categories, Int. Math. Res. Not. 2008 (2008), Art. ID rnn029, 17 pages, arXiv:0709.0882.
  10. Garver A., McConville T., Oriented flip graphs and noncrossing tree partitions, arXiv:1604.06009.
  11. Keller B., Vossieck D., Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), 239-253.
  12. Manneville T., Pilaud V., Geometric realizations of the accordion complex of a dissection, arXiv:1703.09953.
  13. Palu Y., Pilaud V., Plamondon P.-G., Non-kissing complexes and tau-tilting for gentle algebras, arXiv:1707.07574.
  14. Stasheff J.D., Homotopy associativity of $H$-spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275-292.
  15. Stasheff J.D., Homotopy associativity of $H$-spaces. II, Trans. Amer. Math. Soc. 108 (1963), 293-312.
  16. Tamari D., Monoï des préordonnés et chaî nes de Malcev, Ph.D. Thesis, Université de Paris, 1951.

Previous article  Next article   Contents of Volume 14 (2018)