A GL_n(q)-analogue of the partition lattice

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 Cohen-Macaulay, 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 q-analogue. The proof of the Cohen-Macaulay 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.


This talk will be preceded by a VIGRE pre-talk, 3:15-3:45pm, in 3866 East Hall. The pre-talk will very briefly discuss shellability of simplicial complexes and its consequences, then introduce topics from matroid theory such as shellability of matroid complexes, internal activity, and the Tutte polynomial. This pre-talk will attempt to serve as a continuation of the pre-talk on matroid theory that took place earlier in the semester, without assuming all attendees necessarily were at that pre-talk. The poset PD_n(q) to be discussed in the seminar talk is not known to be shellable, but rather its homotopy type is computed using discrete Morse theory. However, it may be useful to compare the result for PD_n(q) with the way in which matroid complexes are shellable.