A generalization of the dual immaculate quasisymmetric functions in partially commutative variables

TL;DR

This paper introduces a new pair of dual bases in colored quasisymmetric and noncommutative symmetric functions, extending the immaculate bases with combinatorial and algebraic properties, including expansion rules and Hopf structure.

Contribution

It generalizes the immaculate and dual immaculate bases to colored algebras, providing combinatorial definitions, expansion formulas, and Hopf algebra structure insights.

Findings

Defined colored dual immaculate functions via tableaux

Established expansion rules and Pieri rule for colored immaculate functions

Extended methods to colored row-strict immaculate functions

Abstract

We define a new pair of dual bases that generalize the immaculate and dual immaculate bases to the colored algebras $QSym_A$ and $NSym_A$. The colored dual immaculate functions are defined combinatorially via tableaux, and we present results on their Hopf algebra structure, expansions to and from other bases, and skew functions. For the colored immaculate functions, defined using creation operators, we study expansions to and from other bases and provide a right Pieri rule. This includes a combinatorial method for expanding colored immaculate functions into the colored ribbon basis that specializes to a new analogous result in the uncolored case. We use the same methods to define colored generalizations of the row-strict immaculate and row-strict dual immaculate functions with similar results.