Structure Constants for Immaculate Functions

TL;DR

This paper explores the structure constants of immaculate functions, providing new proofs, properties, and counterexamples related to their algebraic behavior, akin to classical symmetric function theory.

Contribution

It offers a new proof of the Pieri rule, establishes translation invariance, and presents a counterexample to an analogue of the saturation conjecture for immaculate functions.

Findings

New proof of the left Pieri rule for immaculate functions

Translation invariance property of structure coefficients

Counterexample to the analogue of the saturation conjecture

Abstract

The immaculate functions, $\mathfrak{S}_{\alpha}$, were introduced as a Schur-like basis for $\operatorname{\mathsf{Nsym}}$. 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.