The University of Michigan Combinatorics Seminar
Winter 2003
January 31, 4:10-5:00, 3866 East Hall





Analogues of the Lie representation for classical groups

John Shareshian

Washington University




Abstract

The symmetric group Sn acts on the lattice Pin of partitions of [n] into subsets in the obvious way, and this action determines a representation of Sn on the unique nontrivial homology group of the order complex of Pin. Results of Hanlon and Stanley show that this representation is obtained by tensoring with the alternating representation the representation of Sn 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 Sn. I will discuss my efforts to find analogous represenations for GLn(q) and other classical finite groups.