February 27, 1997
Lynne Butler, Haverford College and MSRI
"Enumeratively identical lattices:
Subgroups and invariant subspaces"
Subgroups of a finite abelian group G, ordered by inclusion, are the
elements of a partially ordered set that may be studied with a wide
variety of combinatorial techniques (including the algebra of symmetric
functions and the combinatorial theory of shellable posets). If the
isomorphism type of G is specified by a partition of n, then this
poset shares enumerative properties with the poset of T-invariant
subspaces of the vector space of dimension n over the field with
p elements, where T is a nilpotent transformation whose Jordan blocks
have sizes the parts of the partition.
This talk compares and contrasts these two posets. In particular, we
discuss the recent result obtained with Karl that these two posets are
isomorphic if the third largest part of the partition is at most one.
To illustrate the classical result that these two posets are otherwise
non-isomorphic, we have spectacular pictures created with Burgiel of
the case when p=2 and the partition is 222.