University of Georgia VIGRE
Algebra Group
Our research group was initiated in August 2003 under the Vertical
Integration of Research and Education (VIGRE)
Program sponsored by the National Science Foundation (NSF) at the
Department of Mathematics at the University
of Georgia (UGA). The group was initially led from 2003-05
by David Benson, Brian Boe and Daniel Nakano.
The faculty leaders from 2005-08 were Brian Boe, Leonard Chastkofsky,
and Daniel Nakano. In 2010-11, the
group will be led by Brian Boe, Jon Carlson, Leonard Chastkofsky, and
Daniel Nakano.
The VIGRE Algebra Group consists of faculty members, postdoctoral
fellows, graduate students and undergraduates. Our group
operates much like a laboratory in the physical and biological sciences
by bringing individuals with varying backgrounds and expertise
to work on a common research project. One of our main educational
objectives has been to develop innovative methods for conducting
mathematical research in a large group atmosphere.
During the 2003-04 academic year we focused our efforts on
understanding conjugacy classes of nilpotent orbits
via the use of cohomology and representation theory. In 2004-05, our
group studied support varieties for modules
over the symmetric group. From 2005-08, we investigated methods
for computing cohomology for Lie algebras, quantum
groups and other related objects. Our group has completed seven papers
including two software packages:
- Varieties for Nilpotent Matrices over Simple Lie Algebras I:
Good Primes, J. Algebra 280, (2004), 719-737.
- Varieties for Nilpotent Matrices over Simple Lie Algebras II:
Bad Primes, J. Algebra 292, (2005), 65-99.
- Support Varieties for Weyl Modules over Bad Primes, J. Algebra
312, (2007), 602-633.
- On Kostant's Theorem for Lie Algebra Cohomology, Cont. Math. 478,
(2009), 39-60.
- An Analog of Kostant's Theorem for the Cohomology of Quantum
Groups, Proceedings of the AMS 138, (2010), 85-99.
- First Cohomology for Finite Groups of Lie Type: simple modules
with small dominant weights, arXiv:1010.1203, to appear in the
Transactions of the AMS.
- Second Cohomology for Finite Groups of Lie Type, submitted.
- GAP program UGA3, companion to the paper "Support varieties for
Weyl modules over bad primes", (gzipped tar or
zip format)
to
determine support varieties of induced modules for exceptional groups
over
bad primes.
- GAP code for calculations used in the papers on "First and Second
Cohomology for Finite Groups of Lie Type": UGAVIGRE6-7code
- A minor correction to a table at the top of page 732 in the paper
"Varieties for
Nilpotent Matrices over Simple Lie Algebras I: Good Primes": Revised Table
GAP is available from http://www-gap.dcs.st-and.ac.uk/~gap/. Versions
are available for UNIX/Linux, Windows, or Macintosh OS.
From 2009-11, we will investigate the cohomology of finite groups.
In particular we will focus our attention on computations of
low degree cohomology for symmetric groups and finite groups of Lie
type.
The details about how our research was carried out in a large
group setting can be found in our biannual reports:
Fall
2003 Report
Spring
2004 Report
Fall
2004 Report
Spring
2005 Report
Fall
2005 Report
Spring
2006 Report
Fall 2006 Report
Spring 2007 Report
Fall 2007 Report
Spring 2008 Report
Fall 2009 Report
Spring 2010 Report
Fall 2010 Report
Spring 2011 Report
Other mathematicians have found the questions that we have worked on
interesting and our results useful. Here are some papers where our
group's work has been cited.
- J.F. Carlson, Z. Lin, D.K. Nakano, Support varieties for modules
over Chevalley groups and classical Lie algebras, Trans. AMS, 360,
(2008), 1879-1906.
- I. Bagci, J.R. Kujawa, D.K. Nakano, Cohomology and support
varieties for Lie superalgebras of type W(n), IMRN, (2008), Art. Id
rnn115.
- A.G. Elashvili, V.G. Kac, On exceptional nilpotents in semisimple
Lie algebras, J. Lie Theory, 19, (2009), 371-390.
- D.J. Hemmer, The complexity of certain Specht modules for the
symmetric group, J. Alg. Comb., 30, (2009), 421-427.
- K.J. Lim, The varieties of some Specht modules, J. Algebra, 321,
(2009), 2287-2301.
- K.J. Lim, The complexity of the Specht modules corresponding to
hook partitions, Arch. Math, 93, (2009), 11-22.
- B.D. Boe, J.R. Kujawa, D.K. Nakano, Cohomology and support
varieties for Lie superalgebras II, Proc. LMS, 98, (2009), 19-44.
- B.D. Boe, J.R. Kujawa, D.K. Nakano, Cohomology and support
varieties for Lie superalgebras, Trans. AMS, 362, (2010), 6551-6590.
- J. Feldvoss, S. Witherspoon, Support varieties and
representation type of small quantum groups, IMRN, (2010), 1346-1362.
- T. Arakawa, Associated varieties of modules over Kac-Moody
algebras and C_2 finiteness of W-algebras, arXiv1004.1554.
Click here for a for a detailed list of members of the group
(2003-11):
Members
of
the UGA VIGRE Algebra Group