Free algebraic structures on the permutohedra

TL;DR

This paper demonstrates that the vector space of faces of permutohedra naturally forms a free tridendriform algebra and identifies a basis, revealing new algebraic structures related to operads and cacti algebras.

Contribution

It proves the freeness of the tridendriform algebra structure on permutohedra faces and explicitly constructs a basis, advancing the understanding of algebraic structures on polytopes.

Findings

The face vector space of permutohedra forms a free tridendriform algebra.

The subspace of primitive elements is a free cacti algebra.

Explicit basis for the free tridendriform algebra is provided.

Abstract

Tridendriform algebras are a type of associative algebras, introduced independently by F. Chapoton and by J.-L. Loday and the third author, in order to describe operads related to the Stasheff polytopes. The vector space $\st$ spanned by the faces of permutohedra has a natural structure of tridendriform bialgebra, we prove that it is free as a tridendriform algebra and exhibit a basis. Our result implies that the subspace of primitive elements of the coalgebra $\st$ , equipped with the coboundary map of permutohedra, is a free cacti algebra.