Finite Reflection Groups and Linear Preserver Problems

Chi-Kwong Li, Ilya Spitkovsky, Nahum Zobin
2004 Rocky Mountain Journal of Mathematics  
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 .
doi:10.1216/rmjm/1181069902 fatcat:22mcxlvb5vab7met7igtxmrkc4