Majorization and Inequalities among Complete Homogeneous Symmetric Functions
This result settles a decade-old open problem in the theory of symmetric functions, providing a definitive negative answer for higher degrees.
The paper resolves an open question by Cuttler, Greene, and Skandera (2011) on whether majorization characterizes inequalities among complete homogeneous symmetric functions (CHs). It proves that for every degree greater than 7, majorization does not characterize these inequalities, contradicting the pattern observed for degrees up to 7.
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.