The University of Michigan Combinatorics Seminar


Abstract 

We introduce a GL_n(q)analogue of the partition lattice, namely the
lattice of partial direct sum decompositions of a finite vector space,
denoted PD_n(q). Discrete Morse theory and matroid theory are used to
show that PD_n(q) is homotopically CohenMacaulay, implying its homology
is concentrated in top degree, and hence its virtual GL_n(q)Lefschetz
character is the negative of an actual character. This actual character
is shown to be an induced linear character which seems to be the
GL_n(q)analogue of the symmetric group character on the top homology of
the partition lattice. Along the way, the lattice of
partial partitions of a finite set is introduced and shown to be a
collapsible, supersolvable lattice which has PD_n(q) as its qanalogue.
The proof of the CohenMacaulay property for PD_n(q) uses the fact that
PD_n(q) is covered by copies of the
lattice of partial partitions sitting inside, much
like apartments,
with one copy for each basis in the matroid of lines in a finite vector
space. This is joint work with Phil Hanlon and John Shareshian. 