Dual Immaculate Quasisymmetric Functions Expand Positively into Young Quasisymmetric Schur Functions

摘要

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