1.  Introduction:  The course leads to the forefronts of one of the big achievements in 20th century mathematics: Representation Theory develops powerful applications of group theory via linear representations which are fundamental in many parts of mathematics. Algebra I and II are prerequisites and Group Theory is a good accompaniment which is best taken first or concurrently.

2.  Timetable Hours, Credits, Assessments:  The course is a 20 UCU unit of 33 lectures, 5 problem sheets and occasional problem sessions. Assessment is by marked homework (20%) and examination (80%).

3.  Overview:  Group theory has many applications in science, especially in physics and crystallography. Most of these applications are realised via representation theory. This course provides an introduction to the main ideas and notions of this powerful theory, explains some the machinery and formulates its basic results. Representation Theory demonstrates the enormous power of linear algebra and builds upon what students have learned at level 1 and 2 in Pure Mathematics I and II and Algebra I and II. Essentially, this topic is a symbiosis of group theory and linear algebra. This feature is important for understanding mathematics as a whole rather than as a union of disjoint theories.

The course starts by introducing the main notions and showing a variety of examples. The idea of equivalence of representations is crucial for the theory, although initially quite difficult to grasp. After this introduction some of the profound and classical results of the theory such as Schur's Lemma and Maschke's Theorem are proved. Their role in the topic cannot be overestimated. The so-called Second Schur Lemma is the key to the theory. Introducing the notions of group average and the Euclidean metric on the vector space of group functions allows us to deduce the orthogonality relations for matrix entries of group representations. One of the most powerful and efficient methods in representation theory is the theory of group characters. This exposition occupies a considerable part of the course. The theory of induced representations introduces students to efficient and practical methods of constructing and analysing representations. The course ends with a variety of examples.

The Advanced Topic will deal with the representation theory of the symmetric groups. This subject is an important part of group theory and has many applications in mathematics and science more generally.

4.  Recommended Reading:  All of the following should be in the Library. There are many other books in the library that cover the material of the course.

Vector space background. Vector space complements. Projectors onto a subspace. Properties of the trace of a matrix. (1 lecture)

Inner product spaces. Orthogonality of vectors and subspaces. Orthogonal complement. (2 lectures)

Review of group theoretical background: Abelian groups and generators. Homomorphisms. Isomorphisms. General linear group. (2 lectures)

Linear groups. Reducible linear groups, matrix interpretation. Complete reducibility, matrix interpretation. Block-triangular and block-diagonal linear groups. Maschke's theorem. (3 lectures)

Irreducible linear groups, Schur's lemma. Version for algebraically closed fields. (2 lectures)

Linear and matrix representations. Examples. Representations of a group of order 2. (1 lecture)

Direct sum of representations. Interpretation of Maschke's theorem for representations. (1 lecture)

Irreducible representations. Interpretation of Schur's lemma. (1 lecture)

Restriction of a representation of a group G to a G-stable subspace. Restriction of a representation of a group G to a subgroup. (1 lecture)

Equivalence of representations. Geometric and matrix form. Examples. (1 lecture)

Triangulation of a representation of an abelian group over an algebraically closed field. (1 lecture)

Schur's second lemma. Group average map: . Corollaries. (2 lectures)

The space of functions on a finite groups. Natural basis, dimension. (1 lectures)

Orthogonality relations for matrix entries of a representation. Finiteness of the number of non-equivalent irreducible representations. (2 lectures)

 Characters

Characters of representations. Elementary properties. Characters of equivalent representations. (1 lecture)

Irreducible characters of cyclic groups. (1 lecture)

Orthogonality relations for characters of representations. Refinement for complex representations. (2 lectures)

The space of class functions on a group. Basis of irreducible characters. The number of non-equivalent irreducible representations. (2 lectures)

The regular representation and its character. Expression of the group order as the sum of squares of dimensions of non-equivalent irreducible representations. (2 lectures)

Application of orthogonality relations for characters. Evaluations of inner products of characters. Evaluations of multiplicity of an irreducible character in an arbitrary one. (2 lectures)

Induced representations. The character of an induced representation. (1 lectures)

The representation theory of the symmetric group over the complex numbers.