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Abstract: Let P be a highly symmetric polytope on which a group G acts by automorphisms, and let K be a G-invariant CW subcomplex of P. The action of G on K induces an action of G on the homology groups of K. We will look at situations where (a) a basis for the homology of K can be constructed from the combinatorics of P using discrete Morse theory, and (b) the complex characters of the homology representations may be explicitly described. We will concentrate on the polytopes known as the half cube and hypersimplex, and outline points of contact with other seemingly unrelated problems. This work is joint with Jacob Harper and is supported by NSF grant DMS-0905768.