Editing Root system (section)

==The root poset==
[[File:E6HassePoset.svg|class=skin-invert-image|thumb|300px|[[Hasse diagram]] of E6 [[Root system#The root poset|root poset]] with edge labels identifying the added simple root]]
The set of positive roots is naturally ordered by saying that <math>\alpha \leq \beta</math> if and only if <math>\beta-\alpha</math> is a nonnegative linear combination of simple roots. This [[Partially ordered set|poset]] is [[Graded poset|graded]] by <math display="inline">\deg\left(\sum_{\alpha \in \Delta} \lambda_\alpha \alpha\right) = \sum_{\alpha \in \Delta}\lambda_\alpha</math>, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset.<ref>{{harvnb|Humphreys|1992|loc=Theorem 3.20}}</ref> The Hasse graph is a visualization of the ordering of the root poset.