Varieties of modules and representations of Frobenius kernels of reductive groups
Lead Research Organisation:
Lancaster University
Department Name: Mathematics and Statistics
Abstract
An algebra A is a type of mathematical object with a structure satisfying certain properties; a representation for A is a space on which A acts in a way which is compatible with its structure. Varieties of modules arise when we consider the set of *all* possible representations (of a given dimension) for A. Beginning with elementary examples, one obtains surprisingly rich and complex geometric structures parametrizing the n-dimensional modules.
One natural question to ask is the following: given an algebra A, can we identify the irreducible components of the variety of n-dimensional A-modules? In general, this turns out to be a hard question. The existing methods for tackling it mostly depend on fairly restrictive properties of the algebra A. In the research proposed here, we will investigate varieties of modules for a particular class of algebras: group algebras of elementary abelian p-groups of rank 2. The problem of describing these varieties of modules has an alternative interpretation in relation to cohomology of the second Frobenius kernel of the group of invertible n x n matrices, due to work of Suslin, Friedlander and Bendel. In order to tackle this specific problem, we will have to develop some new methods for studying varieties of modules, adapting earlier results of Crawley-Boevey and Schroer.
One natural question to ask is the following: given an algebra A, can we identify the irreducible components of the variety of n-dimensional A-modules? In general, this turns out to be a hard question. The existing methods for tackling it mostly depend on fairly restrictive properties of the algebra A. In the research proposed here, we will investigate varieties of modules for a particular class of algebras: group algebras of elementary abelian p-groups of rank 2. The problem of describing these varieties of modules has an alternative interpretation in relation to cohomology of the second Frobenius kernel of the group of invertible n x n matrices, due to work of Suslin, Friedlander and Bendel. In order to tackle this specific problem, we will have to develop some new methods for studying varieties of modules, adapting earlier results of Crawley-Boevey and Schroer.
Planned Impact
This research is in fundamental mathematics, and so is likely to be of most interest to fellow academics. We will carry out some calculations in cohomology of modules for elementary abelian p-groups; if any of our methods might be of general interest, we will publish the computer codes online.
Organisations
People |
ORCID iD |
| Paul Levy (Principal Investigator) |
Publications
Fu B
(2017)
Generic singularities of nilpotent orbit closures
in Advances in Mathematics
Fu Baohua
(2017)
Generic singularities of nilpotent orbit closures
in ADVANCES IN MATHEMATICS
Levy P
(2022)
Commuting varieties and cohomological complexity theory
in Journal of the London Mathematical Society
Ngo N
(2014)
On varieties of commuting nilpotent matrices
in Linear Algebra and its Applications
Ngo N
(2015)
Corrigendum to "On nilpotent commuting varieties and cohomology of Frobenius kernels" [J. Algebra 425 (2015) 65-84]
in Journal of Algebra
Ngo N
(2015)
On nilpotent commuting varieties and cohomology of Frobenius kernels
in Journal of Algebra
| Description | This is research in pure mathematics and it is therefore not possible to explain it to a non-specialist audience. (1) In one project, the dimension of the r-th nilpotent commuting variety of a Lie algebra of type A or C was determined for all r sufficiently large. This was joint work of the PI (Paul Levy), the Senior Research Associate who worked on the project (Nham Ngo) and a collaborator working in Slovenia (Klemen Sivic). (2) A second project involved the determination of the irreducible components of certain so-called "mixed (nilpotent) commuting varieties". The most complete results are determined for the exceptional Lie algebras. (3) Thirdly, the PI was able to complete the publication of a paper on singularities in nilpotent cones of exceptional Lie algebras (jointly with Fu, Juteau and Sommers). |
| Exploitation Route | The results on nilpotent commuting varieties contribute to the study of cohomological support varieties for finite group schemes, especially the Frobenius kernels of reductive groups. While support varieties were described in general terms in the 1990s, very little concrete information has been available for the "higher" Frobenius kernels. On the other hand, in other contexts the theory of support has been extremely useful in representation theory (of e.g. restricted Lie algebras). The results in (1) and (2) provide some general information about the properties of support varieties for the higher Frobenius kernels and so will contribute to the support variety scenery for these group schemes. The research in (3) fits into a broader area of research on the connections between symplectic singularities and simple Lie algebras. One outcome of our work was the discovery of hitherto unknown isolated symplectic singularities, which are likely to be of interest to specialists in complex geometry. Some subsequent research, building on our results, will also shed light on the structure of the so-called "special pieces" which are defined by means of representation theory of reflection groups. |
| Sectors | Other |
| Title | Jordan types |
| Description | I wrote a GAP subroutine which allows one to determine the Jordan type of an element of any nilpotent orbit in an exceptional Lie algebra over a field of positive characteristic. Here the Jordan type refers to the action on the adjoint or the minimal faithful module. A subset of this data was known (e.g. by work of "the VIGRE group" in the US) but it was produced by rather more laborious methods. In practice, once the calculation has been carried out, one can just look at the relevant table rather than re-doing the computation. Most of the results obtained will be published in due course. |
| Type Of Material | Computer model/algorithm |
| Year Produced | 2016 |
| Provided To Others? | Yes |
| Impact | This would only ever have academic beneficiaries, and that only once the tables have been published. |