For natural families of polytopes determined by substructures (e.g., tours or matchings) of a complete structure (e.g., undirected graph, directed graph, or uniform hypergraph), we show how D.G. Higman's theory of coherent configurations can be used to establish the spectra (and hence the dimension) of classes of faces induced by other substructures (e.g., cliques) of the structure. Furthermore, we show how the algebra can be used to derive a maximal irredundant set of equations satisfied by every point of such a face. We work out complete details of the proof machinery when the structure is the complete undirected graph and the faces are induced by cliques. As an example we consider the subtour elimination faces of the Hamiltonian tour polytope. Our approach extends the methodology introduced by J. Lee for determining the dimension of some combinatorially described polytopes.