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Prospectus

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Representation Theory (BM)

Course
2021-2022

Admission requirements

Linear Algebra 1 and 2, Algebra 1.
Recommended: Algebra 2.

Description

Representation theory is about understanding and exploiting symmetry using linear algebra. The central objects of study are linear actions of groups on vector spaces. This gives rise to a very structured and beautiful theory, which can be viewed as a generalisation of Fourier analysis to a non-commutative setting.

Representation theory plays a major role in mathematics and physics. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles.

After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. 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.

Course Objectives

  1. Basics in ring theory
  2. Basics in category theory
  3. Basics in homological algebra, Wedderburn-Artin, Maschke’s theorem
  4. Group representations, characters, character tables
  5. Burnside’s theorem

Mode of instruction

Lectures and homework

Assessment method

Homework (considered as ‘practical = praktische oefening’), written final exam, and retake.
In the following, h is the averaged homework grade with the lowest two grades dropped, f is the
grade for the final exam and r is the grade for the retake.
Final grades are rounded to the nearest allowable grade, and rounded up when there are 2
nearest allowable grades.
Final grade after final exam: if f is at least 5 then 0.2h + 0.8f, and otherwise f.
Final grade after retake: max(0.2h +0.8r, r) if r is at least 5, and otherwise r.

Recommended literature

Pavel Etingof, Introduction to Representation Theory. American Mathematical Society, 2011.
Hendrik Lenstra: Representatietheorie, 2003. Lecture notes by Jeanine Daems and Willem Jan Palenstijn (in Dutch). There is an English translation by Gabriele Dalla Torre.

Contact

edix@math.leidenuniv.nl