Symmetric association schemes arising from abstract regular polytopes


This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schläfli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.