Non-leaving-face property for marked surfaces

doi:10.1007/s11464-019-0767-7

Front. Math. China

2019, Vol. 14

Issue (3) : 521-534 https://doi.org/10.1007/s11464-019-0767-7

RESEARCH ARTICLE

Non-leaving-face property for marked surfaces

Thomas BRÜSTLE¹, Jie ZHANG²(

)

¹. Département de Mathématiques, Université de Sherbrooke, Sherbrooke, J1K 2R1, Canada
². School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

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Abstract

We consider the polytope arising from a marked surface by flips of triangulations. D. D. Sleator, R. E. Tarjan, and W. P. Thurston [J. Amer. Math. Soc., 1988, 1(3): 647{681] studied the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations. They showed that every shortest path between two vertices in a face does not leave that face. We give a new method, which is different from the one used by V. Disarlo and H. Parlier [arXiv: 1411.4285] to establish the same non-leaving-face property for all unpunctured marked surfaces.

Keywords Marked surface non-leaving-face property exchange graph

Corresponding Author(s): Jie ZHANG

Issue Date: 10 July 2019

Cite this article:

Thomas BRÜSTLE,Jie ZHANG. Non-leaving-face property for marked surfaces[J]. Front. Math. China, 2019, 14(3): 521-534.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0767-7
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I3/521

1	T Brüstle, Y Qiu. Tagged mapping class groups I: Auslander-Reiten translation. Math Z, 2015, 279(3): 1103–1120 https://doi.org/10.1007/s00209-015-1405-z
2	T Brüstle , D Yang. Ordered exchange graphs. In: Benson D J, Krause H, Skowro?nski A, eds. Advances in Representation Theory of Algebras. EMS Ser Congr Rep. Z?urich: Eur Math Soc, 2013, 135–193 https://doi.org/10.4171/125-1/5
3	T Brüstle, J Zhang. On the cluster category of a marked surface without punctures. Algebra Number Theory, 2011, 5(4): 529–566 https://doi.org/10.2140/ant.2011.5.529
4	T, Brüstle J Zhang. A module-theoretic interpretation of Schiffler's expansion formula. Comm Algebra, 2013, 41(1): 260–283 https://doi.org/10.1080/00927872.2011.629267
5	A B Buan, R Marsh, M Reineke, I Reiten, G Todorov. Tilting theory and cluster combinatorics. Adv Math, 2006, 204(2): 572–618 https://doi.org/10.1016/j.aim.2005.06.003
6	C Ceballos, V Pilaud. The diameter of type D associahedra and the non-leaving-face property. European J Combin, 2016, 51: 109–124
7	F Chapoton, S Fomin, A Zelevinsky. Polytopal realizations of generalized associahedra. Canad Math Bull, 2002, 45(4): 537–566 https://doi.org/10.4153/CMB-2002-054-1
8	V Disarlo, H Parlier. The geometry of flip graphs and mapping class groups. arXiv: 1411.4285
9	S Fomin, M Shapiro, D Thurston. Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math, 2008, 201(1): 83–146 https://doi.org/10.1007/s11511-008-0030-7
10	S Fomin, A Zelevinsky. Cluster algebras. I. Foundations. J Amer Math Soc, 2002, 15(2): 497–529 https://doi.org/10.1090/S0894-0347-01-00385-X
11	S Fomin, A Zelevinsky. Cluster algebras. II. Finite type classification. Invent Math, 2003, 154(1): 63–121 https://doi.org/10.1007/s00222-003-0302-y
12	C Hohlweg, C E M C Lange, H Thomas. Permutahedra and generalized associahedra. Adv Math, 2011, 226(1): 608–640 https://doi.org/10.1016/j.aim.2010.07.005
13	D Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces. Proc Lond Math Soc (3), 2009, 98(3): 797–839 https://doi.org/10.1112/plms/pdn051
14	H Parlier, L Pournin. Once punctured disks, non-convex polygons, and pointihedra. arXiv: 1602.04576
15	H Parlier, L Pournin. Flip-graph moduli spaces of filling surfaces. J Eur Math Soc (JEMS), 2017, 19(9): 2697–2737 https://doi.org/10.4171/JEMS/726
16	H Parlier, L Pournin. Modular flip-graphs of one-holed surfaces. European J Combin, 2018, 67: 158–173 https://doi.org/10.1016/j.ejc.2017.07.003
17	L Pournin. The diameter of associahedra. Adv Math, 2014, 259: 13–42 https://doi.org/10.1016/j.aim.2014.02.035
18	N Reading. Cambrian lattices. Adv Math, 2006, 205(2): 313–353 https://doi.org/10.1016/j.aim.2005.07.010
19	D D Sleator, R E Tarjan, W P Thurston. Rotation distance, triangulations, and hyperbolic geometry. J Amer Math Soc, 1988, 1(3): 647–681 https://doi.org/10.2307/1990951
20	N Williams. W-associahedra have the non-leaving-face property. European J Combin, 2017, 62: 272–285 https://doi.org/10.1016/j.ejc.2017.01.006

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