Editing Root system (section)

==Classification of root systems by Dynkin diagrams==
{{See also|Dynkin diagram}}
[[File:Finite Dynkin diagrams.svg|class=skin-invert-image|480px|thumb|Pictures of all the connected Dynkin diagrams]]

A root system is irreducible if it cannot be partitioned into the union of two proper subsets <math>\Phi=\Phi_1\cup\Phi_2</math>, such that <math>(\alpha,\beta)=0</math> for all <math>\alpha\in\Phi_1</math> and <math>\beta\in\Phi_2</math> .

Irreducible root systems [[bijection|correspond]] to certain [[Graph (discrete mathematics)|graphs]], the '''[[Dynkin diagram]]s''' named after [[Eugene Dynkin]]. The classification of these graphs is a simple matter of [[combinatorics]], and induces a classification of irreducible root systems.

===Constructing the Dynkin diagram===

Given a root system, select a set Δ of [[root system#Positive roots and simple roots|simple roots]] as in the preceding section. The vertices of the associated Dynkin diagram correspond to the roots in Δ.  Edges are drawn between vertices as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.)
*No edge if the vectors are orthogonal,
*An undirected single edge if they make an angle of 120 degrees,
*A directed double edge if they make an angle of 135 degrees, and 
*A directed triple edge if they make an angle of 150 degrees.  
The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.)

Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, <math>\sqrt 2</math>, <math>\sqrt 3</math>. In the case of the <math>G_2</math> root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of <math>\sqrt 3</math>). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.)

===Classifying root systems===
Although a given root system has more than one possible set of simple roots, the [[Weyl group]] acts transitively on such choices.<ref>This follows from {{harvnb|Hall|2015|loc=Proposition 8.23}}</ref> Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same.<ref>{{harvnb|Hall|2015|loc=Proposition 8.32}}</ref>

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. A root systems is irreducible if and only if its Dynkin diagram is connected.<ref>{{harvnb|Hall|2015|loc=Proposition 8.23}}</ref> The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

If <math>\Phi</math> is a root system, the Dynkin diagram for the dual root system <math>\Phi^\vee</math> is obtained from the Dynkin diagram of <math>\Phi</math> by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that <math>B_n</math> and <math>C_n</math> are dual to each other.