Description
One of the most monumental results in all of mathematics is the
classification of all finite simple groups (i.e., groups with no
proper non-trivial normal subgroups): apart from easy groups like
cyclic groups of prime order and alternating groups, there are also
26 sporadic groups and the groups of 'Lie type'. The simple groups
of Lie type are essentially matrix groups (e.g.,
\(\mathrm{PSL}_n(\mathbb{F}_q)\) over finite fields \(\mathbb{F}_q\))
and form the most important class of finite simple groups.
In the project, we will study how to construct the finite groups of
Lie type, which include the simple ones, in a way which reveals
their geometric structure. The second part of the project will be an
introduction to the representation theory of finite groups of Lie
type. This centres around the construction of Deligne and Lusztig of
representations using techniques of algebraic geometry/topology
(cohomology).
In Michaelmas we will cover:
- A very brief introduction to algebraic varieties and
algebraic groups. An algebraic variety is essentially the
set of solutions to a system of polynomial equations over a
field. An algebraic group is essentially an algebraic variety
with a group structure. Alad
- Reductive groups over finite fields. We define
so-called reductive algebraic groups. A finite group of Lie type
is a reductive group whose points are taken in a finite field.
Alternatively, they are the fixed points, under a so-called
Frobenius endomorphism, of a reductive group defined over a
finite field.
- Subgroup structure. Finite groups of Lie type have a
nice and rich structure. We will look at maximal tori, Borel
subgroups, unipotent subgroups, parabolic subgroups, Weyl
groups. All of these have simple interpretations in matrix
groups like \(\mathrm{GL}_n\) or \(\mathrm{SL}_n\), and we will look
at these examples, together with others like symplectic groups.
Using Borel subgroups and the so-called Bruhat decomposition, we
will be able to count the number of elements in finite groups of
Lie type.
At some point in Epiphany, we will look at Deligne-Lusztig
varieties and representations and some of their properties, as
well as examples for small groups.
Prerequisites
You will need Representation Theory IV as a co-requisite.
Resources
- R. Carter, Introduction to algebraic groups and Lie
algebras, in Carter and Geck Representations of reductive
groups.
- F. Digne and J. Michel, Representations of Finite Groups
of Lie Type.
- M. Geck, An Introduction to Algebraic Geometry and
Algebraic Groups,
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