Majorization and Inequalities among Complete Homogeneous Symmetric Functions
Jia Xu, Yong Yao
TL;DR
This work addresses whether majorization completely characterizes inequalities among complete homogeneous symmetric functions (CHs), a question posed after known results for other bases. The authors prove that for every degree $d\ge 8$, the majorization order fails to capture all CH inequalities, by constructing witness partitions $\mu,\lambda$ with $\mu$ not majorizing $\lambda$ yet $H_{n,\mu}\ge H_{n,\lambda}$ for all $n$. The proof hinges on an explicit $d=8$ base case proved via induction on the standard simplex and a relaxation method to extend the result to higher degrees. This establishes a fundamental difference between CHs and other symmetric function families and invites search for a relaxed or alternative characterization of CH inequalities.
Abstract
Inequalities among symmetric functions are fundamental in various branches of mathematics, thus motivating a systematic study of their structure. Majorization has been shown to characterize inequalities among commonly used symmetric functions, except for complete homogeneous symmetric functions (shortened as CHs). In 2011, Cuttler, Greene, and Skandera posed a natural question: Can majorization also characterize inequalities among CHs? Their work demonstrated that majorization characterizes inequalities among CHs up to degree 7 and suggested exploring its validity for higher degrees. In this paper, we show that, for every degree greater than 7, majorization does not characterize inequalities among CHs.