Basic Representation Theory, Fall 2026

Basic Representation Theory

Vladimir Ivanov

Syllabus of the course

Lecture 1 notes – The Notion of a Representation, Problem Set 1

Lecture 2 notes – Schur’s Lemma. Maschke’s Theorem, Problem Set 2

Lecture 3 notes – Decomposition of a Representation. One-dimensional Representations, Problem Set 3

Lecture 4 notes – Tensor Product of Representations, Problem Set 4

Lecture 5 notes – Character of a Representation

Recommended textbooks

Representation Theory.

1. W. Fulton, J. Harris, Representation theory. A first course., Corrected edition, 2004 (Chapters 1, 2, 4, 7, 8, 11, 12)
2. E.B. Vinberg, Linear Representations of Groups, 1st edition, 2nd printing, 2010.
3. G.James, M.Liebeck, Representations and Characters of Groups., 2nd edition, 2001 (For the first half of the course, Sections 1-20)
4. B. Hall, Lie Groups, Lie Algebras, and Representations. An Elementary Introduction., 2nd edition, 2015 (For the second half of the course, Chapters 1-4)

Reviewing Linear Algebra.

1. S. Freidberg, A. Insel, L. Spence, Linear Algebra. (The sections without an asterisk)
2. S. Axler, Linear Algebra Done Right.
3. S. Roman, Advanced Linear Algebra (Chapters 1-3, 7-11, 14, 18).
4. The lecture notes of my Advanced Linear Algebra course.

Reviewing Group Theory.

1. C.Pinter, A book of abstract algebra. (Chapters 1-16)
2. J.Fraleigh, A first course in abstract algebra. (Sections 1-14)

Reviewing Multivariable Calculus and introductory Differentiable Manifolds notions (for the second half of the course).

1. W.Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry (Chapters 1-3)
2. J.Lee, Introduction to Smooth Manifolds (for a deeper and more general picture, Chapters 1-3)

Additional resources.

1. Youtube channel of professor Richard E. Borcherds
2. Youtube channel of professor Steven Roman