Basic Representation Theory, Fall 2025

Basic Representation Theory

Vladimir Ivanov

Syllabus of the course

Preliminary discussion 1 – Vector Spaces, Bases

Preliminary discussion 2 – Linear Maps, Matrices, Dual Spaces

Preliminary discussion 3 – Tensor Products of Vector Spaces

Preliminary discussion 4 – Jordan Canonical Form Theorem for Complex Linear Operators

Preliminary discussion 5 – Complexification

Preliminary discussion 6 – Inner Product Spaces

Preliminary discussion 7 – Operators in Inner Product Spaces

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 – Complexification, Problem Set 4

Lecture 5 notes – Character of a Representation, Problem Set 5

Lecture 6 notes – Orthogonality Relations, Problem Set 6

Lecture 7 notes – Group Algebra, Problem Set 7

Lecture 8 notes – Matrix Lie Groups, Problem Set 8

Lecture 9 notes – Lie Algebras, Problem Set 9

Lecture 10 notes – Lie Groups and Lie Algebras Representations, Problem Set 10

Lecture 11 notes – Lie Groups and Lie Algebras Representations, part 2, Problem Set 11

Lecture 12 notes – Irreducible representations of the Lie algebra sl_3(C)

Lecture 13 notes – Irreducible Complex Characters of Symmetric Groups

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).

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