Editing Root system (section)

===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.