The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals
Abstract
This paper introduces the geometric foundations for the study of the -permutahedron and the -associahedron, two objects that encode the underlying geometric structure of the -weak order and the -Tamari lattice. We introduce the -permutahedron as the complex of pure intervals of the -weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the -associahedron as the complex of pure -Tamari intervals of the -Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen -associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.
Keywords
Cite
@article{arxiv.2309.14261,
title = {The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals},
author = {Cesar Ceballos and Viviane Pons},
journal= {arXiv preprint arXiv:2309.14261},
year = {2025}
}
Comments
45 pages, 28 figures