Editing Weight (representation theory) (section)

===Partial ordering on the space of weights===
[[File:Illustration_of_notion_of_"higher"_for_root_systems.png|thumb|right|If the positive roots are <math>\alpha_1</math>, <math>\alpha_2</math>, and <math>\alpha_3</math>, the shaded region is the set of points higher than <math>\lambda</math>]]
We now introduce a partial ordering on the set of weights, which will be used to formulate the theorem of the highest weight describing the representations of <math>\mathfrak g</math>. Recall that ''R'' is the set of roots; we now fix a set <math>R^+</math> of [[Root system#Positive roots and simple roots|positive roots]].

Consider two elements <math>\mu</math> and <math>\lambda</math> of  <math>\mathfrak h_0</math>. We are mainly interested in the case where <math>\mu</math> and <math>\lambda</math> are integral, but this assumption is not necessary to the definition we are about to introduce. We then say that <math>\mu</math> is '''higher''' than <math>\lambda</math>, which we write as <math>\mu\succeq\lambda</math>, if <math>\mu-\lambda</math> is expressible as a linear combination of positive roots with non-negative real coefficients.<ref>{{harvnb|Hall|2015}} Definition 8.39</ref> This means, roughly, that "higher" means in the directions of the positive roots. We equivalently say that <math>\lambda</math> is "lower" than <math>\mu</math>, which we write as <math>\lambda\preceq\mu</math>.

This is only a ''partial'' ordering; it can easily happen that <math>\mu</math> is neither higher nor lower than <math>\lambda</math>.