Surjections as Double Posets

Loïc Foissy ; Frédéric Patras

Séminaire lotharingien de combinatoire, Tome 82 (2020-2021)

Résumé

The theory of double posets and pictures between them, introduced by Malvenuto and Reutenauer, is a far reaching development of Zelevinsky's theory of pictures in that, among others, it embeds the latter into a self-dual Hopf algebraic framework. It has brought forward many ideas and results and has led recently to several developments, in various directions. Namely, besides algebraic combinatorics and noncommutative representation theory: algebraic topology and the geometry of polytopes.

One of these developments, by the first author of this article, was the combinatorial and Hopf algebraic study of symmetric groups from the point of view of double posets. The present article extends these results to surjections. We introduce first a family of double posets, packed double posets. Using an appropriate statistics on surjections that generalizes inversions, it is shown that they are in bijection with surjections or, equivalently, with packed words. The following sections investigate their self-dual Hopf algebraic properties. Using an appropriate notion of linear extensions of packed double posets, the Hopf algebra of packed double posets is proved to be isomorphic with (two different versions of) the Hopf algebra of word quasi-symmetric functions.