Necessary Conditions for Schur-Maximality

Abstract

McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that B ≤ s A if s A − s B is Schurpositive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length a or (a + 1) for a ≥ 2. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal.Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length a being at the end, or on rows of length (a + 1) being evenly distributed. Contents2010 Mathematics Subject Classification. Primary 05E05; Secondary 05E10, 06A05, 06A06, 20C30.

Classifying the near-equality of ribbon Schur functions

“…Now Lemma 3.15 and Lemma 3.17 allow us to identify specific partitions at which the LR coefficients for α and β differ. One immediate consequence is the following necessary condition for Schur-positivity of a difference r α − r β , which generalizes [28,Theorem 40].…”

“…Billera, Thomas, and van Willigenburg have classified when two ribbon Schur functions are equal [3], providing insight towards a combinatorial classification of equality of skew Schur functions [20,24]. Necessary and sufficient conditions have been found for the difference of two ribbon Schur functions to be Schur-positive [16,18,19,28] and the sets of nonzero coefficients in the Schur function expansion are fairly well understood [12,21].…”