The University of Michigan Combinatorics Seminar


Abstract 

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 ncycle
in S_{n}. I will discuss my efforts to find analogous
represenations for GL_{n}(q) and other classical finite
groups.
