Realizations of regular polytopes, III

Abstract

There is a comprehensive theory of geometric realizations of a finite abstract regular polytope ${\mathcal{P}}$ in euclidean spaces. Identifying congruent realizations of ${\mathcal{P}}$, their space forms a pointed convex cone, with scalar multiplication and blending the operations which combine different realizations. In this paper, a new way to combine realizations is introduced, that of the (tensor) product. Just as blending corresponds to sums of representations of groups, so the product corresponds to the usual product of representations. A range of examples is given to illustrate the new theory. As well as being of intrinsic interest, some of these examples lead to extra insight into already known realization spaces; more importantly, the structure of the realization cone of the regular 600-cell is here determined for the first time.