Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)

TL;DR

This paper provides a detailed structural analysis of the Malvenuto-Reutenauer Hopf algebra of permutations, including explicit formulas, primitive elements, and its decomposition, revealing deep connections with combinatorial and algebraic structures.

Contribution

It offers explicit formulas for the antipode, proves the algebra is cofree, and describes its decomposition and structure constants, advancing understanding of this Hopf algebra's structure.

Findings

Explicit antipode formulas derived

Proved the algebra is cofree coalgebra

Connected the algebra's structure to the weak order on symmetric groups

Abstract

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. We also describe the structure constants of the multiplication as a certain number of facets of the permutahedron. Our results reveal a close relationship between the structure of this Hopf algebra and the weak order on the symmetric groups.