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](http://en.wikipedia.org/wiki/Muirhead%27s_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 $$\frac{s \_{\lambda}(x \_1,x \_2,\dots,x \_n)}{s \_{\lambda}(1,1,\dots,1)}\geq \frac{s \_{\mu}(x \_1,x \_2,\dots,x \_n)}{s \_{\mu}(1,1,\dots,1)}?$$
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If not, does majorization induce some similarly nice order on values of Schur polynomials?



Majorization and Schur Polynomials
This question is listed as a conjecture (conjecture 7.4 in the section "Open questions") in a recent [paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-526CFXB-3&_user=10&_coverDate=02%2F17%2F2011&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f28d84462f1b0537df2951b5fd44de42&searchtype=a) of Cuttler, Greene and Skandera.


