Plethysm and the algebra of uniform block permutations

The Hopf Algebra of Uniform Block Permutations

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Ros...

The Hopf Algebra of Uniform Block Permutations. Extended Abstract

Abstract. We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Ge...

A polynomial realization of the Hopf algebra of uniform block permutations

We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters. Résumé. Nous étudions l’algèbre de Hopf combinatoire dont les bases sont indexées par les permutations de blocs uniformes et nous réalisons cette algèbre en termes de polynômes non-commutatifs en une infinité de bi-le...

Hopf Algebra of Permutations

Hopf Algebra of Permutations

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 multiplicati...