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Journal of Combinatorial Algebra


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Volume 3, Issue 4, 2019, pp. 401–438
DOI: 10.4171/JCA/35

Published online: 2019-12-16

Non-kissing and non-crossing complexes for locally gentle algebras

Yann Palu[1], Vincent Pilaud[2] and Pierre-Guy Plamondon[3]

(1) Université de Picardie Jules Verne, Amiens, France
(2) École Polytechnique, Palaiseau, France
(3) Université Paris-Sud, Orsay, France

Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to -tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.

Keywords: Associahedra, non-kissing complex, accordions, non-crossing complex, gentle algebras, tau-tilting

Palu Yann, Pilaud Vincent, Plamondon Pierre-Guy: Non-kissing and non-crossing complexes for locally gentle algebras. J. Comb. Algebra 3 (2019), 401-438. doi: 10.4171/JCA/35