Structure Constants for Immaculate Functions

Abstract

The immaculate functions, ${\mathfrak{S}}_{\alpha }$, were introduced as a Schur-like basis for $\mathrm{Nsym}$. We investigate facts about their structure constants. These are analogues of Littlewood-Richardson coefficents. We will give a new proof of the left Pieri rule for the ${\mathfrak{S}}_{\alpha }$, a translation invariance property for the structure coefficients of the ${\mathfrak{S}}_{\alpha }$, and a counterexample to an ${\mathfrak{S}}_{\alpha }$-analogue of the saturation conjecture.

arXiv comment17 pages

References in corpus (4)

• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions
• Multiplicative structures of the immaculate basis of non-commutative symmetric functions
• The Pieri rule for dual immaculate quasi-symmetric functions

Cited by in corpus (3)

• Multiplicative structures of the immaculate basis of non-commutative symmetric functions
• Theta maps for combinatorial Hopf algebras
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables