Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
Algebraic Topology
2023-06-13 v2
Abstract
In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis. Furthermore, we give counter-examples to the converse.
Cite
@article{arxiv.1811.11770,
title = {Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis},
author = {Joan Bellier-Millès and Bérénice Delcroix-Oger and Eric Hoffbeck},
journal= {arXiv preprint arXiv:1811.11770},
year = {2023}
}
Comments
In this version, we have removed the part on a potential EL-labelling on the poset associated to the operad Perm because of an error reported to us by Rafael Gonz\'alez, Josh Hallam and Yeison Quiceno