A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group.

One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers. Along the way, we will also meet categories, modules and tensor products.

The mark for the exam must be at least 5. Pavel Etingof, Introduction to Representation Theory. American Mathematical Society, Freely available on the author's website.

## MATH / Linear Representations of Finite Groups

Serge Lang, Algebra. Springer-Verlag, Hendrik Lenstra, Representatietheorie. Lecture notes in Dutch. Zoek naar Vakken Opleidingen Keywords Academic year Zoeken. Representation theory of finite groups BM Vak.

### Description

Mention of Brauer's Theorem on induced representations. The Odd order theorem: Every group of odd order is solvable -- a massive generalization of Burnside's theorem, very difficult! The orthogonal group O 2 as a "continuous" dihedral group i.

- Representation theory.
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We get a 2-dimensional faithful representation of the group, which must have a limit point in the compact O 2. Get the dual or contragredient of a representation. The space of G-equivariant maps between two finite dimensional irreducible representations over an algebraically closed field is either:. These will give us powerful tools to construct nontrivial representations from ones that we already have.

Andreas has already described how tensor products work on a basis. I will describe the universal property that is behind them. This scalar is the determinant of T. In this way, we see how to define det T without choosing a basis of V. It makes sense that we should be able to do this, since we know a posteriori that the determinant doesn't depend on the choice of basis. Assume G is finite and char F is not divisible by card G.

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Averaging over group, which makes an arbitrary linear map between G-representations into a G-equivariant map. Every character is a class function. In the other direction, we want to understand a general class function in terms of characters. We are in the process of proving this. We have almost completely proved 2. Note: 2. We already know it is linearly independent from 2. After we complete the proof that the irreducible characters are orthonormal, we need to show that they span CF G.

As a consequence, we get that the sum of the squares of the dimensions of the irreducible characters is the order of G. Examples of induction: Permutation representations, sum of inductions is the induction of the sums, inducing a subrepresentation, interaction with tensor product projection formula.

Assuming their is a k-sums of squares identity, we deduced the Hurwitz matrix equations and a k-dimensional representation of a hypothetical group G. Topic outline General.

Room: in House 6 both lectures and exercise sessions. First meeting: 18 January. First meeting 29 January, week 5. Check the schedule! Thus [Se, 2.

### Topic outline

Grades : Assessed based on ca. Material : The first part of the course will cover Chap. Some goals: Define representations. Study basic operations and properties of representations. Two key theorems: 1. The isomorphism type of a finite-dimensional complex representation of a finite group is uniquely determined by its trace called the character of the representations.