Finite reflection groups

The finite groups generated by reflections (g. g. r. ) of real Euclidean space of n dimensions (") have been classified by Coxeter 4. He has noticed a number of properties common to these groups, but has been able to prove them only by verification in the individual cases. Our prime purpose here is to give general proofs of some of these results (1. 1 to 1. 4 below). If is a finite g. g. r. on En, the reflecting hyperplanes (r. h. ) all pass through one point, which may be taken as the origin 0, and partition E" into a number of chambers each of which is a fundamental region of ; further is generated by the reflections in the walls of any one of these chambers. The group is irreducible in the usual algebraic sense if and only if there are n linearly independent r. h. and there is no partition of the r. h. into two nonempty sets which are orthogonal to each other 7, p. 403. In this case each chamber is a simplicial cone with vertex at 0 3, p. 254; 4, p. 590. This leads us to the first result of Coxeter 4, p. 610: If is a finite irreducible g. g. r. on En and if h is the order of the product of the reflections in the walls of one of the fundamental chambers, then the number of reflecting hyperplanes is nh/2. Associated with each simple Lie algebra (or Lie group) of rank n over the complex field there is a finite irreducible g. g. r. on " and a set of vectors (roots) normal to the corresponding r. h. l; 13. There then exists a fundamental set of roots and a so-called dominant root relative to this set (definitions in 6 and 8). Then Coxeter's second observation 6, p. 234 is this: 1. 2. Theorem. If ax, a2, , an is a fundamental set of roots for a simple Lie algebra of rank n, and if ʸ'a, is the dominant root, then the number of reflecting hyperplanes of the corresponding group (or one-half the number of roots) is w (l + X) yO/2. From 1. 2 (see 6, p. 212) one immediately gets: 1. 3. Theorem. The dimension of the Lie algebra (or Lie group) isn (2+ XO- As Coxeter 6, p. 212 has remarked, this is an interesting analogue to the formula of Weyl for the order of, namely, g=f-n\ T>, with /-1 denoting the number of y's equal to 1.