Topics in Representation Theory

Let $V$ be a finite dimensional vector space. If $G$ is a group, then a representation is a group homomorphism $\rho: G \rightarrow \operatorname{GL}(V)$. Each element $G$ corresponds to some invertible linear transformation. This induces an action, of the group $G$, on the vector space, where we write $g \cdot v = \rho(g)(v)$. Using this action, we say that $V$ is a $G$-module.

Mathematicians try to classify the objects that they are studying. It is always nice if we can understand how and why two entities are different. At least one reason for this, is to take something complicated and identify it with something that we understand very well. One of the major accomplishments of the twentieth century was the classification of finite simple groups. Representation theory played a big role in this work. The classification is broken down into:

1. Finite cyclic abelian groups
2. The alternating groups
3. Simple groups of Lie Type
4. Sporadic Groups

One way that representation theory manifests itself naturally is in the study of symmetries. One of the first finite groups one encouters in an abstract algebra course is that fo the dihedral groups. These are the symmetries of regular polygons. If one envisions a square in the plane, the rotation of 90 degrees is a symmetry of the square. Another symmetry is that of a reflection. These are both natural ideas that are studied to some extent in High School geometry. It turns out that the square has eight symmetries, and can be realized by the rotation of 90 degrees, a fixed reflection, and all combinations of these two elements. \[ D_8 = \langle r, s \vert r^4 = s^2 = 1, srs = r^{-1} \rangle \] This presentation tells us that if we rotate the square four times we get back to where we started, and the same if we reflect the square twice. The last relation tells us that the rotation $r$ and the reflection $s$ do not commute. That is $rs \neq sr$. There is another presentation of this group using only reflections. Here is how the dihedral group of order eight is presented as a Coxeter group \[ D_8 = \langle \alpha, \beta \vert \alpha^2 = \beta^2 = (\alpha \beta)^4 = 1 \rangle \]

These ideas arise in both chemistry and physics. Chemists use representations to study the structure and symmetries of crystals for instance. Physics researchers are interested in representations in the study of particle physics.

Modular Representation Theory: One of the goals of representation theory is to describe all of the indecomposable representations, of a finite group $G$. The indecomposables are the building blocks of the representation. The easiest ones to understand are the simple representations. An introduction to the representation theore of finite groups is usually done over the complex numbers. Most of these methods can be extended to the situation where the characteristic of the field does not divide the order of the group. This is the so-called Ordinary Representation Theory. Maschke's theorem says that every ordinary representation can be decomposed into simples.

Modular representation theory strives to understand the case when the characteristic of the field divides the order of the group. The following results hold in ordinary representation theory, and all fail in the modular case:

1. Maschke's Theorem (The decomposition into simples.)
2. Hochster - Eagon Theorem (The ring of invariants is Cohen-Macaulay)
3. The surjectivity of the transfer homomorphism

Quiver Theory: A quiver $Q$ is a directed graph and a representation of $Q$ is where a vector space is attached to each vertex and a linear mapping to each arrow. This seemingly simple idea leads to very deep results. An expository article by Derksen and Weyman can be found here. It turns out that every finite-dimesnional basic associative algebra can be realized as a path algebra $\operatorname{mod}$ some ideal. A result of Drozd says that every finite-dimensional associative algebra is either of:

1. Finite Type
2. Tame Type
3. Wild Type

I found the study of quivers and finite-dimesional algebras quite compelling. My enjoyment stemmed from the fact that som many areas of mathematics come together in the problems that arise. Quiver theory involves results from algebraic combinatorics, algebraic geometry, commutative algebra, geometric invariant theory, and homological algebra. It was in a course in quiver theory that I got my introduction to geometric invariant theory.

Here are some notes, from a series of talks, that I gave on modular representation theory:

• Notes regarding the basic properties of algebras: Properties of Algebras
• A good dose of quiver theory: Quiver Theory
• Character Theory in Modular Representations: Character Theory

References:

1. Alperin, J.L., Local Representation Theory: Modular Representations as an introduction to the Local Representation Theory of Finite Groups , Revised edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1993.
2. Benson, D.J., Modular Representation Theory: New Trends and Methods , Second corrected printing, Lecture Notes in Mathematics, Springer, 2007.
3. Erdmann, Karin, Blocks of Tame Representation Type and Related Algebras , Lecture Notes in Mathematics, Springer, 1990
4. Lux, Klaus, Pahlings, Herbert, Representations of Groups: A Computational Approach , Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010.
5. Neusel, Mara, D., Smith, Larry, Invariant Theory of Finite Groups , Mathematical Surveys and Monographs, American Mathematical Society, 2010
6. Serre, Jean-Pierre, Linear Representations of Finite Groups , Fifth corrected edition, Graduate Text in Mathematics, Springer, 1996.