Inequalities among Symmetric Polynomial Functions: Counter-examples and New Conjectures

Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene and Skandera in 2011, on inequalities among the complete homogeneous symmetric polynomial function $H_{n,\lambda }$: It states that the inequality $H_{n,\lambda }\le H_{n,\mu }$ implies majorization order $\lambda ⪯\mu$. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for number of variables $n=3$ and degree $d=8$. They then asked whether the conjecture is true when $n$ is sufficiently large. In this paper, we show, by a family of counter-examples, that the conjecture does not hold for any $n$ and any $d$ as long as $n\ge 2$ and $d\ge 8$. Based on the insights gained from the counter-examples, we propose a new conjecture for the inequality $H_{n,\lambda }\le H_{n,\mu }$.