We determine all cases where double coset algebras of string Coxeter groups with respect to maximal parabolic subgroups have the involutive double coset property. For infinite affine string Coxeter groups, this property was only known to occur in the case of the vertex stabilizer for the types ˜C2 and ˜G 2, our new work adds the infinite Coxeter groups of type ˜Cn(n > 2) and ˜F4 to the list. Finite homomorphic images of these groups include the automorphism groups of abstract regular polytopes. An application of the above result is that polytopes of these types will always have polyhedral realization cones. We also investigate the algebraic structure of these double coset algebras from the association scheme perspective. We give an overview of their orders, ranks, commutativity, and character tables, investigate their nontrivial closed subsets, quotients and fusions, and look for possible relationships between the association schemes corresponding to various types. The data we generate leads us to make some new conjectures relating types B and D that we are able to confirm in some cases.

A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics, University of Regina. vi, 72 p.