# Representation theory

MATH-334 / **5 credits**

**Teacher: **

**Language:** English

**Remark:** pas donné en 2021-22

## Summary

Study the basics of representation theory of groups and associative algebras.

## Content

Representations of associative algebras in complex vector spaces. General results: subrepresentations and semisimple representations. Characters. The Jordan-Hölder and Krull-Schmidt theorems. Examples of algebras, quiver algebras and universal enveloping of Lie algebras.

Representations of finite groups. Maschke's theorem. Character theory, orthogonality relations. Burnside's theorem. Induced reprensentations. Frobenius reciprocity.

Representations of the symmetric groups over C. Young diagrams, Young tableaux. Specht modules. Schur-Weyl duality.

## Keywords

Linear representation, subrepresentation, quotient, semisimple representation, character of a representation, associative algebra, representation of a finite group, character table, orthogonality relations, induced representation, restricted representation, symmetric group, Young diagram.

## Learning Prerequisites

## Required courses

Linear algebra or Advanced Linear algebra; Group theory

## Recommended courses

Lie algebras, Coxeter groups

## Learning Outcomes

By the end of the course, the student must be able to:

- Apply concepts and ideas of the course
- Reason rigorously using the notions of the course
- Choose an appropriate method to solve problems
- Identify the concepts relevant to each problem
- Apply concepts to solve problems similar to the questions in problem sets
- Solve new problems using the ideas of the course
- Implement appropriate methods to study and construct representations of groups and algebras

## Teaching methods

Lectures and exercise sessions

## Assessment methods

One take-home written assignment (15% of the grade)

Written exam (85% of the grade)

## Supervision

Office hours | No |

Assistants | Yes |

Forum | No |

## Resources

## Bibliography

1. P. Etingof, O. Goldberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, "Introduction to Representation Theory". Student Mathematical Library Volume: 59; 2011. ISBN: 978-0-8218-5351-1

2. Fulton, William, and Joe Harris. *Representation Theory: A First Course*. Graduate texts in mathematics. Vol. 129. New York, NY: Springer, 1991. ISBN: 9780387974958.

## Ressources en bibliothèque

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Representation theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

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