The The VIGRE Algebra Group was led by David Benson (during January), Brian Boe, and Daniel Nakano.
The other members of the group are

Faculty:

Leonard Chastkofsky

Postdocs:

Nadia Mazza
Jonathan Kujawa

Graduate Students:

Irfan Bagci
Phil Bergonio
Bobbe Cooper
Sarah Hofmann
Kenyon J. Platt
Dong Hoon Shin
Stephen Winburn
Caroline Wright


The main goal of the VIGRE Algebra Seminar this term was to continue the study of support varieties for modules over
the symmetric group begun last Fall. In the first three weeks, Benson and Nakano gave lectures reviewing what we had done last
semester, and quickly bringing the new students up to speed. At the end of Fall we had started to compute the supports for Specht
modules for the symmetric group on d letters when d < p^2 (where the characterstic of the underlying field is p). Early this term we proved
a conjecture giving the dimension of the support varieties in this setting.

Most of this semester was focused on the situation d is p^2 or larger. We wrote a program in MAGMA to compute the restriction
of the support variety to elementary abelian subgroups, where it manifests itself as ``rank varieties'' which completely determine its
structure. The students generated data for a large number of examples, and we made a conjecture for the ``generic'' behavior in the
principal block.  Mazza, Kujawa, and our former postdoc David Hemmer gave lectures on some tools that might help to prove the conjecture.
Several sessions were spent refining the conjecture, and trying to understand the ``non-generic'' partitions in the principal block as well as the
situation for other blocks. Enhancements to the MAGMA program included looking at branching, decomposability, and socle filtrations. All of the
students made significant progress in understanding representation theory and cohomology.

In the course of disseminating the results of our previous work on support varieties for Weyl modules over bad primes, Platt and Wright
revisited the GAP program they had helped write last summer, streamlining the code and adding documentation. The program was posted on the
VIGRE Algebra website so that, for instance, 3rd parties can check the computational aspects of our results.

In the future we hope to formulate a general conjecture on the dimensions of the support varieties for Specht modules in arbitrary blocks, and
find tools to prove it.