MATH 506 - Group Representation Theory (Spring 2026)

| General Information | Homework | Lecture Notes | Other Resources |

This is a graduate course in representation theory. The most important prerequisite (by far) is a first course in graduate algebra (Math 500).

A representation is a (linear) action by a group on a vector space. From this straightforward definition, we get a large variety of phenomena which relate to almost every area of mathematics.

The course plan (subject to change) is as follows. We will start with an overview of the representation theory of finite groups, moving into symmetric groups. We will then consider representations of Lie groups and Lie algebras (prior knowledge of Lie theory is not assumed). Finally, we will cover some number of miscellanous topics.

General Information

Lecture	MWF 10:00am-10:50am Noyes Laboratory 164
Instructor	Andrew Hardt Office: Harker Hall 204C Email: ahardt@illinois.edu Office Hours: MF 11:00am-11:50am, W 2:00pm-2:50pm
More Details	Syllabus | Gradescope | If you can't access the Gradescope course, email me

Homework

Homework assignments will be roughly biweekly, generally due at 9am via Gradescope. Using LaTeX for your homework is strongly encouraged.

• Homework 1 (TeX file) -- due Wed. 2/11 at 9am (Solutions)
• Homework 2 (TeX file) -- due Wed. 2/25 at 9am (Solutions)
• Homework 3 (TeX file) -- due Wed. 3/25 at 9am

Lecture Notes

• Lecture 1 (1/26): Overview, examples
• Lecture 2 (1/28): Definitions, basic examples
• Lecture 3 (1/30): Schur's Lemma, Maschke's Theorem
• Lecture 4 (2/2): Intro to character theory
• Lecture 5 (2/4): Character orthogonality and decomposition of the regular representation
• Lecture 6 (2/6): Column orthogonality and character table examples
• Lecture 7 (2/9): Restriction and induction, Frobenius reciprocity
• Lecture 8 (2/11): Partitions and tableaux, Young subgroups
• Lecture 9 (~2/13): Representations of GL_2(F_q) (Video Recording)
• Lecture 10 (2/16): Tabloids, polytabloids, and Specht modules
• Lecture 11 (2/18): Irreducibility of Specht modules and triangularity of M^lambda decomposition
• Lecture 12 (2/20): Irreducibility of Specht modules and triangularity of M^lambda decomposition (cont.)
• Lecture 13 (2/23): Basis for Specht modules
• Lecture 14 (2/25): Basis for Specht modules (cont.), branching rule
• Lecture 15 (2/27): Branching rule (cont.), decomposition of M^mu
• Lecture 16 (3/2): Decomposition of M^mu (cont.)
• Lecture 17 (3/4): Decomposition of M^mu (cont.), statement of Murnaghan-Nakayama rule
• Lecture 18 (3/6): sl_2(C) and its representations
• Lecture 19 (3/9): Root systems
• Lecture 20 (3/11): Classification of root systems

Textbooks and Other Resources

Course Resources

These are the sources we're following most closely in lecture.

• Fulton and Harris, Representation Theory, A First Course
• Serre, Linear Representations of Finite Groups
• Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions
• Bump, Lie Groups
• Humphreys, Introduction to Lie Algebras and Representation Theory
• Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

Other Representation Theory Material

Other books and notes you might find useful.

• Webb, A Course in Finite Group Representation Theory
• Etingof et al., Introduction to representation theory
• Teleman Lecture Notes
• Speyer Lecture Notes
• Piatetski-Shapiro Complex Representations of GL(2, K) for Finite Fields K
• James The Representation Theory of the Symmetric Groups
• Geck and Pfeiffer Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras
• Gooman and Wallach Symmetry, Representations, and Invariants

Helpful Links

• Overleaf, a cloud-based LaTeX editor
• Detexify, a web app for identifying the LaTeX command(s) for various symbols
• Guidelines for Good Mathematical Writing, by Francis Su

Bonus Links

Useful, relevant, and/or fun. Feel free to send me anything interesting you run across.