Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis

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 Poincaré-Birkhoff-Witt basis. Furthermore, we give counter-examples to the converse.

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 González, Josh Hallam and Yeison Quiceno