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

Faculty:

Leonard Chastkofsky

Postdocs:

Markus Hunziker
Jo Jang Hyun
Jonathan Kujawa
Nadia Mazza

Graduate Students:

Phil Bergonio
Bobbe Cooper
Jerry Hower
Graham Matthews
Kenyon Platt
Caroline Wright

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 of two types. The first type consists
of generalizations of the ``restricted nullcone'' which manifests itself in the study of support varieties for restricted Lie
algebras. The second type consists of the support varieties themselves.

In Fall 2003, the group determined explicitly (as a union of orbit closures) the first class of varieties -- 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 than the original proof of Carlson, Lin, Nakano, and
Parshall, the computation of the restricted nullcone (the case r=p). This work has been submitted for publication.

In Spring 2004, we solved the same problem over fields of bad characteristic. Even for the restricted nullcone, the results
are new. As a corollary, we answered affirmatively a 1986 question of Friedlander-Parshall, as to whether the restricted
nullcone is always an irreducible variety. We also determined explicitly the analogous varieties of order-r unipotent elements
in the group. We discovered a previously unknown bijection, preserving dimensions and inclusions among closures,
between the nilpotent orbits in the restricted nullcone and the unipotent classes in the ``restricted unipotent variety'' (the case r=p).
No such bijection can exist between the full nilpotent and unipotent varieties in bad characteristic, because the numbers of
nilpotent and unipotent orbits are not the same.

For the exceptional groups, the ingredients necessary for our unipotent variety computations were available in the literature,
but not for the nilpotent orbits. We first had to determine representatives for the orbits; in some cases even these were not in
the literature. Nakano did most of the work of producing representatives. Then Graham Matthews, with assistance from Markus
Hunziker and Nadia Mazza, wrote a program using the computer algebra software MAGMA to compute the necessary Jordan block
sizes. Jonathan Kujawa and Caroline Wright entered the representatives and generated the data. To complete the determination
of the order-r unipotent and nilpotent varieties, we broke the large group into smaller groups led by Benson, Boe and Chastkofsky.
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.

For the classical groups, Boe did much of the work of assembling, assimilating, and presenting to the group the ingredients from the
literature, with assistance from Benson and Nakano.  Three group members, Phil Bergonio, Jerry Hower, and Jo Jang Hyun,
were given the task of determining the order-r nilpotent varieties. They made conjectures, which were subsequently proved by Hower.

We now have a second paper which is almost complete, in which we determine explicitly (as a union of orbit closures), by elementary
methods, the order-r nilpotent and unipotent elements for the minimal (and, for the exceptional groups, adjoint) representations, for all
r and all simple Lie algebras in good characteristic. In particular, it includes the computation of the restricted nullcones. The paper also
contains the complete list of nilpotent orbit representatives for the exceptional Lie algebras, the associated Jordan block sizes, and the
Hasse diagrams of orbit closure inclusions inside the restricted nullcone. The graduate students were involved in writing the paper.

In the latter part of the semester, the group has begun work on determining the support varieties for Weyl modules in bad characteristic.
Hunziker and Nakano presented lectures on background for this topic, including results on the solution of the problem in good
characteristic by Nakano, Parshall, and Vella. Chastkofsky wrote a program in Mathematica to calculate some of the relevant data for
Weyl modules. The group again broke into the three smaller subgroups to analyze the data and complete the determination of the support
varieties. The majority of the exceptional cases are now finished. Chastkofsky has begun working on the classical groups; he recently
presented his preliminary results and conjectures. In future work, we hope to completely determine the support varieties of the
Weyl modules for all simple groups in bad characteristic. We will also investigate whether the orbit bijection between the restricted
nilpotent and unipotent varieties arises from an isomorphism between these two varieties themselves.