9 - Unitary Groups and Hermitian Forms

Summary

In the classical theory, the structure of a finite regular polytope or an infinite regular tessellation or honeycomb is governed by a real quadratic form. This form determines the geometry of the ambient space. The symmetry group of the polytope or tessellation is a group of isometries in this space, and is in fact the Coxeter group associated with the quadratic form as in Sections 3A and 3D.

As we shall see in Chapter 11, much of the correspondence between polytopes and forms remains true for an important class of abstract regular polytopes. Here the real quadratic form is replaced by a hermitian form on a finite-dimensional complex space. Moreover, a subgroup of finite index in the automorphism group Γ = Γ(P) of such a polytope P is represented as a group of isometries with respect to this form. In particular, this subgroup (and thus P itself) is finite if and only if the hermitian form is positive definite, in which case it is a finite group generated by reflexions of period 2 in the ambient unitary space.

Unfortunately, we cannot generally be sure that the abstract and geometric groups are actually isomorphic; this may have to be shown on a case-by-case basis.