| Abstract:
The focus of this talk will be on algebraic vector bundles over
Pn and their applications to the Garsia-Haiman representation
theoretic model of the Macdonald two-parameter symmetric polynomials. The
Garsia-Haiman interpretation involves a certain bigraded Sn-module,
Hm, indexed by partitions
m of n. They conjecture that
the dimension of this module is n!.
Using an inductive approach to the dimension conjecture, Bergeron
and Garsia consider the relationship between the module
Hm, for
m a partitions of n+ 1,
and certain predecessor modules,
Hmi,
for mi a partition of n
contained in m. Bergeron
and Garsia formulate conjectures regarding the sums and
intersections of these spaces.
We will give a geometric interpretation of the work of Bergeron and Garsia.
We reinterpret their work within the context of a vector bundle over the Hilbert
scheme of n points in the plane. We will also discuss a general result
concerning the decomposition of vector bundles over Pn into
well-known indecomposable bundles. Using these general results, we will
apply them to the context of the Garsia-Haiman modules. |
alexsuciu@neu.edu