# Integral representations of finite groups and lattice point geometry

This contains both a reference request, and a specific problem. Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider the following property of $\theta(K)$ (not merely of $\theta$): there exists a finite subset $S$ of $H = {\bf Z}^d$ such that

(0) $0 \in S$;

(i) $\cup_{n \geq 1} nS = H$ [here $nS$ means the set of sums of $n$ elements of $S$];

(ii) $S$ is $\theta$-invariant;

(iii) if we set $C$ to be the convex hull (in Euclidean space, ${\bf R}^d$), of $S$, then $0$ belongs to the interior of $C$ and the natural action of $\theta$ on $C$ is transitive on the set of facets of $C$ [a facet is a face of codimension one; the transitivity condition is that every facet can be moved to all other facets by elements of $\theta(K)$];

(iv) for each facet $F$ of $C$, let $S_F =S \cap {\rm cvx } \{F,0\}$; we require that $S_F$ generate $H$ as a group, and $C_F:=\cup_{n\geq 1} nS_F$ is a simplicial cone.

Despite this ridiculously complicated set of conditions, there are plenty of natural examples.

Example 1 Take $K ={\bf Z}_2^d$ acting as the group of diagonal $\pm1$ matrices on ${\bf Z}^d$, and set $S = \{0; \pm e_i \}$ where the $e_i$ vary over the standard basis. Then $C$ is just a $d$-dimensional version of the octohedron, and the facets correspond to the intersection with the orthants, and it is clear that $\theta$ acts transitively on them. The simplicial condition is obvious. [Here, $K$ acts far from transitively on the extreme points of $C$, but this does not matter.]

Example 2 Let $K = \cal S_{d+1}$, the full permutation group on $d+1$ symbols, or any subgroup which acts transitively on the symbols. Then $K$ acts on ${\bf Z}^{d+1}$ by permuting the standard basis elements, and leaves $v= (1,1,\dots, 1)$ invariant. So we obtain the quotient action of $K$ on ${\bf Z}^{d+1}/v{\bf Z}$ identified with ${\bf Z}^{d}$. With $S$ being the orbit of $e_1$ in ${\bf Z}^{d}$ together with $0$, that is, $S = \{0, e_1, e_2,\dots, e_d; -\sum e_i \}$, then it is easy to see that $C$ is a simplex, and the simplicial condition holds, and transitivity is obvious.

Example 3 Let $d = 2$, and let $\theta : K \to {\rm GL}(2,{\bf Z})$ be one to one (faithful). Provided $K$ has more than two elements, then an $S$ satisfying (0--iv) exists.

For the last, the only group which requires more than a few sentences is $K = C_2 \times C_2$, for which there exist two outer conjugacy classes of faithful representions on ${\bf Z}^2$.

The existence of a set $S$ with the properties (0--iv) is obviously an invariant of the integral representation $\theta$ of $K$, in fact an outer invariant: it is an invariant of the image of $K$, $\theta(K) \subset {\rm GL}(d,{\bf Z}^{d})$. This type of thing (using integral geometry to obtain invariants for integral representations) must have been done before.

First question [finally] A reference request for analyzing integral representations of finite groups by lattice point geometry.

Second question Are there more examples than the ones I described, or more generally, can useful necessary/sufficient conditions be derived for the existence of such a set $S$?

Apparently, it is necessary that the trivial representation not be a subrepresentation of $\theta$, and multiplicities should be avoided.

The motivation comes from studying random walks and suitable weight functions on the semidirect product groups ${\bf Z}^{d}\times_{\theta} K$; when such an $S$ exists, the random walks (weight functions) have nicer properties than when no such $S$ exists.