Finite Reflection Groups and Linear Preserver Problems

Let G be one of the Coxeter groups A n , B n , D n , or I 2 (n), naturally acting on a Euclidean space V , and let L(G) stand for the set of linear transformations φ of End V that satisfy φ(G) = G.It is easy to see that L(G) contains all transformations of the form X → P XQ, X → P X * Q, where P, Q belong to the normalizer of G in the orthogonal group and P Q ∈ G.We show that in most cases these transformations exhaust L(G); the only (rather unexpected) exception is the case G = B n .Lemma 1.1.Let P, Q belong to the normalizer N (G) of G in the orthogonal group O(V ), and assume that P Q ∈ G. Then the transformations X → P XQ and X → P X * Q are in L(G).These transformations constitute a group.