The VIGRE Algebra Group was led by David
Benson, Brian Boe, and Daniel Nakano. The other members
of the group are
Faculty:
Leonard Chastkofsky
Jerome Jungster
Postdocs:
Nadia Mazza
Jonathan Kujawa
Graduate Students:
Irfan Bagci
Phil Bergonio
Kenyon J. Platt
Stephen Winburn
Caroline Wright
The main goal of the VIGRE Algebra Seminar this term was to to study
support varieties for modules
over the symmetric group. Support varieties were developed 25 years ago
in the pioneering work of
Alperin and Carlson. Since their introduction, these varieties have
played a central role in the cohomology
theory of finite groups and restricted Lie algebras. Although these
varieties are easy to define once the
finite generation of cohomology is established, they are often very
difficult to compute. One of the main
objectives of this project is to make significant progress in computing
these varieties for important
classes of modules.
At the start of the semester Benson presented four lectures on the
elementary properties of the cohomological
variety theory. This was followed by three lectures by Kujawa on
symmetric group representations. Nakano presented
two lectures on the computation of the support varieties for Young and
permutation modules (recent results with
David Hemmer). During the last four weeks of the term Benson led the
group in making explicit computations of
varieties for symmetric groups. We discovered formulas for
certain Specht modules and have 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). We are optimistic this case will be solved near
the beginning of next semester.
Several students in the group, Bergonio, Platt and Wright, finished up
computations on the support varieties of
Weyl modules over bad primes. This effort was initiated and led over
the summer by Chastkofsky. Boe and Nakano have
compiled the data and have written up the results. A preprint (the
third paper written by our group) is now available.
The first VIGRE Algebra Group paper has now appeared: J. Algebra, 280
(2004), 719-737 and the second
VIGRE Algebra Group paper will appear in the new Journal of Algebra
section devoted to computational algebra.
Bagci and Winburn are new members to our group. They are making good
progress with the material and we are optimistic
they will contribute further in the spring.
VIGRE Algebra Group was led by David Benson,
Brian Boe, and Daniel Nakano. The other members
of the group are
Faculty:
Leonard Chastkofsky
Jerome Jungster
Postdocs:
Jo Jang Hyun
Nadia Mazza
Graduate Students:
Phil Bergonio
Bobbe Cooper
G. Michael Guy
Graham Matthews
Kenyon J. Platt
The main goal of the VIGRE Algebra Seminar was
to study the structures of certain varieties
of nilpotent matrices associated to algebraic
groups and their Lie algebras. These varieties are
generalizations of the ``restricted nullcone''
which manifests itself in the study of support varieties
for restricted Lie algebras. Carlson, Lin, Nakano
and Parshall have recently described the restricted
nullcone for all types over fields of good
characterstic.
Their proof employed the verification of a
conjecture by Jantzen on the support varieties
of Weyl modules which was due to Nakano, Parshall
and Vella.
During the semester material was presented on
basic Lie theory and conjugacy classes along with
methods due to Nakano and Tanisaki on
intersecting
orbit closures. The material was presented by Boe
and Nakano. These methods were then adapted for
the purpose of computing these varieties of nilpotent
matrices for faithful representations of the
simple Lie algebra. For the exceptional Lie algebras, we broke
the large group into smaller groups led by
Benson,
Boe and Chastofsky. The work was then subdivided
and each group was in charge of certain
computations.
This was very successful because the entire group
was engaged and involved in the process, and
led to the completion of the computation of these varieties
for exceptional groups over good characteristic
for the minimal and adjoint representations. The graduate
students were also involved in writing this
section
of our work.
At first, we did not know how to handle the
case
when the groups were classical and the representation
was the adjoint representation. However, after
several weeks of thinking about the problem, we were able
to solve it. Much of the credit for this part
of the project is due to Boe with contributions from Benson and
Nakano. We now have a paper which is almost
complete in which we determine explicitly (as a union of
orbit closures), by elementary methods, the
order-r
nilpotent elements for the minimal and adjoint
representations, for all r and all simple Lie
algebras in good characteristic. As a corollary, we successfully
verified using more elementary methods the
computation
of the restricted nullcone (the case r=p).
The group has begun work on looking at these
varieties
over bad characteristics. Each graduate student went
to the library to look up pertinent literature
on the topic and gave a short talk on their findings in the seminar.
These findings have led us to use available
computer
software to do some of these computations. Markus
Hunziker recently gave a talk in our seminar
on the construction of Chevalley basis. Hunziker, Matthews, and
Mazza are looking at ways to implement the
necessary
calculation on MAGMA. Once this is done, we hope to
be able to compute the restricted nullcone for
bad characterstics (an open problem of interest). In future work,
we hope to find new results on support varieties
for Weyl modules in bad characteristic.