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

By Yann Palu, Vincent Pilaud, Pierre-Guy Plamondon. Journal of combinatorial algebra, 2018

55 citations recorded by the source corpus.

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 $\tau$-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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e260ff5e953f886bd799a306f77c9494c7c80a82

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Citations in the corpus

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, background): “We note that the τ-tilting theory of gentle algebras is the algebraic counterpart of the combinatorics of accordions on surfaces [12, 80, 81].”
A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, background): “e homeomorphism Φ ◦ ϕ satisﬁes the hypotheses of Theorem 4.1. Thus A and A′ are derived equivalent. Remark 5.6. If one starts with a gentle algebras A, the marked surface as constructed in [OPS18] or [PPP18] is given either by a thickening of a ribbon graph or by glueing polygons. Finding the genus ”