Coursebooks

Representation theory

MATH-334

Lecturer(s) :

Lachowska Anna

Language:

English

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:

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.

In the programs

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      Lecture
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  • Lecture in French
  • Lecture in English
  • Lecture in German