Colloquium
Dr. Patricia Hersh, Univ of Michigan
Wednesday, December 17, 2003
2:30-3:30 PM
Lockett 285
A GL_n(q) analogue of the partition lattice and discrete
Morse theory for posets
Robin Forman introduced a discrete version of Morse theory
about 10 years ago as a tool for studying the homotopy
type of simplicial complexes and cell complexes by
producing smaller, simpler-to-understand complexes of
critical cells with much of the same topological structure
as the original complex. In joint work with Eric Babson, it
is shown that the order complex of any finite partially
ordered set with unique minimal and maximal elements has a
discrete Morse function with a relatively small number of
critical cells. In this talk, I will review discrete Morse
theory, then discuss this result for poset order complexes
along with more recent developments, motivations for studying
order complexes of posets, and finally will mention some
applications.
The main application in the talk will be to a GL_n(q) analogue
of the partition lattice that was recently introduced in
joint work with Phil Hanlon and John Shareshian. This "lattice
of partial direct sum decompositions of a finite vector space",
denoted PD_n(q), is shown to be homotopically Cohen-Macaulay,
using discrete Morse theory together with some matroid theory.
Its GL_n(q) character on top homology 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.