Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions

TL;DR

This paper proves a conjecture relating dual immaculate functions to vertex operators using a dendriform algebra structure on quasisymmetric functions, connecting to known algebraic structures.

Contribution

It provides an alternative construction of dual immaculate functions via vertex operators and explores their dendriform algebra structure.

Findings

Proves Zabrocki's conjecture on dual immaculate functions.

Establishes a dendriform structure on QSym.

Connects this structure to known dendriform structures on FQSym and WQSym.

Abstract

The dual immaculate functions are a basis of the ring QSym of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an "immaculate tableau" is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary; but each row has to weakly increase). Dual immaculate functions have been introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties. In this note, we prove a conjecture of Mike Zabrocki which provides an alternative construction for the dual immaculate functions in terms of certain "vertex operators". The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras FQSym and WQSym.