I am trying to characterize all finite groups which can be made from two
4D-real-matrix-represented generators. Though matrix coefficients must be real
(I am avoiding complex coefficients since they effectively double the dimension to 8D),
matrix columns are generally not orthogonal (but, through a transformation, one matrix
could be considered orthogonal).
Given that the two matrices form a finite group, their determinants must be +1 or -1.
If columns were orthogonal for both matrices, I believe I would always be looking at one of the
6 possible polytopes (ignoring reflection-multiplied groups from lower-dimensional polytope groups, e.g., 2D cyclic groups which can be of any length).
Or, will I see more groups if I include non-orthogonal matrices? Conversely, if I see one of the
6 possible polytopes, will the two matrices be simultaneously orthonormally-reducible?
Best Answer
There is a basic trick in the representation theory of finite (or more generally, compact) groups that allows you to assume your matrices are orthogonal. Namely, given any finite subgroup $G\subseteq GL_n(\mathbb{R})$, you can always find an inner product which is preserved by each element of $G$ (so if you change to an orthonormal basis with respect to that inner product, $G$ will become a subgroup of the orthogonal group). The trick is that any positive linear combination of inner products is an inner product, so let $\langle\cdot,\cdot\rangle$ be any inner product and define $$\langle x,y\rangle'=\sum_{g\in G}\langle gx,gy\rangle.$$ Then $\langle\cdot,\cdot\rangle'$ is an inner product, and is $G$-invariant because applying an element of $G$ will just permute the summands.