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Geometric Realizations of the Accordion Complex of a Dissection

  • Thibault Manneville
  • Vincent Pilaud
Article
  • 20 Downloads

Abstract

Consider 2n points on the unit circle and a reference dissection \({\mathrm {D}}_\circ \) of the convex hull of the odd points. The accordion complex of \({\mathrm {D}}_\circ \) is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of \({\mathrm {D}}_\circ \). In particular, this complex is an associahedron when \({\mathrm {D}}_\circ \) is a triangulation and a Stokes complex when \({\mathrm {D}}_\circ \) is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection \({\mathrm {D}}_\circ \), generalizing known constructions arising from cluster algebras.

Keywords

Permutahedra Zonotopes Associahedra g-, c- and d-Vectors 

Mathematics Subject Classification

52B11 52B12 13F60 

Notes

Acknowledgements

We thank C. Hohlweg and S. Stella for many helpful discussions on realizations of the associahedron [27] which were the starting point of this paper. We are grateful to F. Chapoton for various conversations on quadrangulations and Stokes posets, and to A. Garver and T. McConville for introducing us with the accordion complexes during FPSAC’16. Their works [9, 20] were highly inspiring and motivating. We also thank N. Thiery for a question which led to the approach of Sect. 5.2, and to P.-G. Plamondon for discussions on the generalization to cluster algebras presented in Sect. 5.3. Finally, we are grateful to two anonymous referees for their attentive reading and their suggestions on the content and presentation which largely improved our original draft.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LIX, École PolytechniquePalaiseauFrance
  2. 2.CNRS & LIX, École PolytechniquePalaiseauFrance

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