The symmetric group S_n acts on the lattice Pi_n of partitions of [n] into subsets in the obvious way, and this action determines a representation of S_n on the unique nontrivial homology group of the order complex of Pi_n. Results of Hanlon and Stanley show that this representation is obtained by tensoring with the alternating representation the representation of S_n on the subspace of the Free Lie algebra on n variables generated by Lie words in which each varibale appears exactly once (thus the name Lie representation). In addition, the Lie represenation is induced from a faithful representation of the subgroup generated by any n-cycle in S_n. I will discuss my efforts to find analogous represenations for GL_n(q) and other classical finite groups.