Notes and Exercises from Humphrey’s

The big theorem of this section gives us that a finite reflection group, call it W , is generated by simple reflections (reflections of simple roots for a given simple system). We also get a useful corollary that, given a simple system the W orbit of any root meets this simple system. Exercise 1 Let Φ be a root system of rank n consisting of unit vectors. If Ψ ⊂ Φ is a set of n roots whose mutual angles agree with those between the roots in some simple system, then Ψ must be a simple system. Solution based on “Jendrik Stelzner” solution on https://math.stackexchange. com/questions/1457907/a-subset-of-roots-whose-mutual-angles-agree-with-those-of-a-simple-system? rq=1.