Dual Immaculate Quasisymmetric Functions Expand Positively into Young Quasisymmetric Schur Functions

TL;DR

This paper introduces a combinatorial formula for decomposing dual immaculate quasisymmetric functions into Young quasisymmetric Schur functions, establishing positivity and symmetry conditions, and providing algorithms for coefficient computation.

Contribution

It presents a new combinatorial formula, proves symmetry conditions, and develops algorithms for decomposing dual immaculate functions into Young quasisymmetric Schur functions.

Findings

Coefficients are given by a combinatorial formula.

Necessary and sufficient conditions for symmetry are established.

Positive expansion of products involving dual immaculate functions is demonstrated.

Abstract

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel-Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases.