On inequalities for normalized Schur functions

TL;DR

This paper proves a conjecture regarding the monotonicity of normalized Schur functions under the dominance order on partitions, providing new insights into inequalities for symmetric functions.

Contribution

It establishes the monotonicity conjecture for normalized Schur functions, offering a novel proof technique potentially applicable to other symmetric functions.

Findings

Proved the conjecture on normalized Schur functions' monotonicity.

Introduced a new proof technique for inequalities in symmetric functions.

Enhanced understanding of inequalities related to symmetric functions.

Abstract

We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M. Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics, 32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions} under the usual (dominance) partial-order on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions.