Finite Unitary Reflection Groups

Introduction. Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors. In the real case we have a group of orthogonal transformations in Euclidean space En. Among such groups the groups generated by reflections have been the object of considerable study (6; 9; 10). The concept of a reflection has recently been extended to unitary space by Shephard (24). A reflection in unitary space is a congruent transformation of finite period that leaves invariant every point of a certain prime, and it is characterised by the property that all but one of the characteristic roots of the matrix of transformation are equal to unity. The remaining root, if the reflection is of period m, is a primitive rath root of unity, and the reflection is then said to be ra-fold. Shephard, in the paper just quoted, has considered a particular class of unitary groups generated by reflections which possess properties closely analogous to those of the real orthogonal groups considered by Coxeter.