Description
One of the most monumental results in 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
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.
In Michaelmas we will cover the following:
- 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.
Topics for further individual study in the latter stages
of the project may include:
- Working through further examples of groups/subgroups
related to classical groups (orthogonal, symplectic).
- Rudiments of counting points of varieties over finite
fields (this leads to number theory and algebraic
geometry).
- Rudiments of Deligne--Lusztig characters (this leads
to the representation theory of finite groups of Lie
type).
Prerequisites
Algebra II. If you want to explore the representation theory
of finite groups of Lie type as your individual further
topic, you 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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