Year: 2025
Type: article
Abstract: Motivated by categorical representation theory, we define the wiggly complex, whose vertices are arcs wiggling around $$n+2$$ n + 2 points on a line, and whose faces are sets of wiggly arcs which are pairwise pointed and non-crossing. The wiggly complex is a $$(2n-1)$$ ( 2 n - 1 ) -dimensional pseudomanifold, whose facets are wiggly pseudotriangulations. We show that wiggly pseudotriangulations are in bijection with wiggly permutations, which are permutations of [2n] avoiding the patterns $$(2j-1) \cdots i \cdots (2j)$$ ( 2 j - 1 ) ⋯ i ⋯ ( 2 j ) for $$i < 2j-1$$ i < 2 j - 1 and $$(2j) \cdots k \cdots (2j-1)$$ ( 2 j ) ⋯ k ⋯ ( 2 j - 1 ) for $$k > 2j$$ k > 2 j . These permutations define the wiggly lattice, an induced sublattice of the weak order. We then prove that the wiggly complex is isomorphic to the boundary complex of the polar of the wigglyhedron, for which we give explicit and simple vertex and facet descriptions. Interestingly, we observe that any Cambrian associahedron is normally equivalent to a well-chosen face of the wigglyhedron. Finally, we recall the correspondence of wiggly arcs with objects in a category, and we develop categorical criteria for a subset of wiggly arcs to form a face of the wiggly complex. Graphical abstract
Source: Mathematische Zeitschrift
Authors: Asilata Bapat, Vincent Pilaud
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G1175115578, G2064873908, G2262748287, G2462415295, Beyond Permutahedra and Associahedra, G2756433129, G2766336479, G3005324705, G300979063, G34346518, G3480869486, G3546882995, G425668148, G4274939422, G4526231891, G4532494212, G4784319644, G5080475149, The geometry of braids and triangulated categories, G6325762137, G69051668, G6910422418, G7013227978, G7084143925, G7264301175, G7304226585, Beyond Permutahedra and Associahedra: Geometry, Combinatorics, Algebra, and Probability, G8295830637, G84477964
OpenAlex ID: W4410241636