A Hopf algebra on permutations with a coupling product

Abstract

We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra $\mathbb{KS}$. In characteristic zero, this implies that $\mathbb{KS}$ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto--Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for $\mathbb{KS}$.

14 pages