# Majorization and Schur Polynomials

Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and $$\lambda_1+\cdots+\lambda_l=\mu_1+\cdots+\mu_l.$$

While playing around with some variation on Muirhead's inequality which says that $$m_{\lambda}(x_1,x_2,\dots,x_n)\geq m_{\mu}(x_1,x_2,\dots,x_n)$$ for all $x_i\geq 0$ when $\lambda \succeq \mu$ (here the $m_{\lambda}$'s are the monomial symmetric polynomials) I ended up investigating if such an inequality holds for Schur polynomials, which are another famous basis of the ring of symmetric functions. I tried a few special cases, and the following seems to hold, but I don't have a proof or a counterexample. So the question is:

Let $x_1,\dots,x_n \geq 0$ and $\lambda\succeq \mu$, is it always true that $$\frac{s _{\lambda}(x _1,x _2,\dots,x _n)}{s _{\lambda}(1,1,\dots,1)}\geq \frac{s _{\mu}(x _1,x _2,\dots,x _n)}{s _{\mu}(1,1,\dots,1)}?$$

If not, does majorization induce some similarly nice order on values of Schur polynomials?

• Perhaps it is possible to introduce a second set of variables, and look at double Schur functions as well... Jul 27 '15 at 0:39

This was also subsequently proved by Rachid Ait-Haddou and Marie-Laurence Mazure in 2018 in FOCM. There's also a variant in a preprint by Apoorva Khare and Terence Tao (see latest version) with "majorization" replaced by "weak majorization" and "all $x_i \geq 0$" replaced by "all $x_i \geq 1$".