Abstract
A quasisymmetric function is assigned to every double poset (that is, every finite set endowed with two partial orders) and any weight function on its ground set. This generalizes well-known objects such as monomial and fundamental quasisymmetric functions, (skew) Schur functions, dual immaculate functions, and quasisymmetric $\left(P, \omega\right)$-partition enumerators. We prove a formula for the antipode of this function that holds under certain conditions (which are satisfied when the second order of the double poset is total, but also in some other cases); this restates (in a way that to us seems more natural) a result by Malvenuto and Reutenauer, but our proof is new and self-contained. We generalize it further to an even more comprehensive setting, where a group acts on the double poset by automorphisms.
Highlights
Double posets and E-partitions have been introduced by Claudia Malvenuto and Christophe Reutenauer [MalReu09]; their goal was to construct a combinatorial Hopf algebra which harbors a noticeable amount of structure, including an analogue of the Littlewood-Richardson rule and a lift of the internal product operation of the Malvenuto-Reutenauer Hopf algebra of permutations
We shall employ these same notions to restate in a simpler form, and reprove in a more elementary fashion, a formula for the antipode in the Hopf algebra QSym of quasisymmetric functions due to Malvenuto and Reutenauer [MalReu[98], Theorem 3.1]
We further generalize this formula to page 2 a setting in which a group acts on the double poset
Summary
Double posets and E-partitions (for E a double poset) have been introduced by Claudia Malvenuto and Christophe Reutenauer [MalReu09]; their goal was to construct a combinatorial Hopf algebra which harbors a noticeable amount of structure, including an analogue of the Littlewood-Richardson rule and a lift of the internal product operation of the Malvenuto-Reutenauer Hopf algebra of permutations. We shall employ these same notions to restate in a simpler form, and reprove in a more elementary fashion, a formula for the antipode in the Hopf algebra QSym of quasisymmetric functions due to (the same) Malvenuto and Reutenauer [MalReu[98], Theorem 3.1]. Page 2 a setting in which a group acts on the double poset (a generalization inspired by Katharina Jochemko’s [Joch13]). In the present version of the paper, some (classical and/or straightforward) proofs are missing or sketched. A more detailed version exists, in which at least a few of these proofs are elaborated on more[1]. A short summary of this paper has been submitted to the FPSAC conference [Grin16b]
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