Editing Root system (section)

==Positive roots and simple roots==
[[File:Base_for_the_G2_root_system.png|class=skin-invert-image|thumb|right|The labeled roots are a set of positive roots for the <math>G_2</math> root system, with <math>\alpha_1</math> and <math>\alpha_2</math> being the simple roots]]
Given a root system <math>\Phi</math> we can always choose (in many ways) a set of '''positive roots'''. This is a subset <math>\Phi^+</math> of <math>\Phi</math> such that 
* For each root <math>\alpha\in\Phi</math> exactly one of the roots <math>\alpha</math>, <math>-\alpha</math> is contained in <math>\Phi^+</math>.
* For any two distinct <math>\alpha, \beta\in \Phi^+</math> such that <math>\alpha+\beta</math> is a root, <math>\alpha+\beta\in\Phi^+</math>.

If a set of positive roots <math>\Phi^+</math> is chosen, elements of <math>-\Phi^+</math> are called '''negative roots'''. A set of positive roots may be constructed by choosing a hyperplane <math>V</math> not containing any root and setting <math>\Phi^+</math> to be all the roots lying on a fixed side of <math>V</math>. Furthermore, every set of positive roots arises in this way.<ref>{{harvnb|Hall|2015|loc=Theorems 8.16 and 8.17}}</ref>

An element of <math>\Phi^+</math> is called a '''simple root''' (also ''fundamental root'') if it cannot be written as the sum of two elements of <math>\Phi^+</math>. (The set of simple roots is also referred to as a '''base''' for <math>\Phi</math>.)  The set <math>\Delta</math> of simple roots is a basis of <math>E</math> with the following additional special properties:<ref>{{harvnb|Hall|2015|loc=Theorem 8.16}}</ref>
*Every root <math>\alpha\in\Phi</math> is a linear combination of elements of <math>\Delta</math> with ''integer'' coefficients.
*For each <math>\alpha\in\Phi</math>, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system <math>\Phi</math> there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.<ref>{{harvnb|Hall|2015|loc=Proposition 8.28}}</ref>