Basis of totally primitive elements of WQSym

TL;DR

This paper constructs an explicit basis for the totally primitive elements of the Word Quasisymmetric Functions Hopf algebra (WQSym) using a new combinatorial family called biplane forests, clarifying its bidendriform structure.

Contribution

It introduces a new combinatorial basis for the totally primitive elements of WQSym, providing the first explicit description of these elements.

Findings

Explicit basis for totally primitive elements constructed

Biplane forests are in bijection with packed words

Primitive elements indexed by biplane trees

Abstract

By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts are 0). In this paper, we construct a basis indexed by a new combinatorial family called biplane forests in bijection with packed words. In this basis, primitive elements are indexed by biplane trees and totally primitive elements by a certain subset of trees. Thus we obtain the first explicit basis for the totally primitive elements of WQSym.