Basic Representation Theory

Representation Theory studies how groups or other algebraic objects act on vector spaces. Representations play a key role in various models in Mathematical Physics (via symmetry groups), Number Theory (via modular forms), Algebraic Combinatorics (via symmetric functions), and other areas of mathematics.

This course is divided into two parts. The first develops the classical theory of complex representations of finite groups. The second introduces matrix Lie groups, Lie algebras, and their representations. Throughout, we focus on finite-dimensional representations.

The course is organized around active, collaborative learning. Several days before each class, you will receive lecture notes containing definitions, examples, and key statements to be proved. Class time will be devoted to discussing your approaches, refining arguments, and addressing questions. The goal is not only to understand the theory, but also to develop fluency in constructing and communicating rigorous proofs.

Prerequisites

Linear Algebra (vector spaces, direct sums, quotient spaces, bases of a vector space, a correspondence between linear maps and matrices, image and kernel of a linear map, dual space, determinant, trace, eigenvectors and eigenvalues, characteristic polynomial, Jordan canonical form theorem over \(\mathbb{C}\), inner product spaces, orthogonal projection, Fourier expansion, spectral theory for unitary, Hermitian, and orthogonal operators), Elementary Theory of Groups (group homomorphisms, quotient groups, first isomorphism theorem, groups defined by generators and relations), Basic point‑set topology (open/closed sets, continuity, compactness in \(\mathbb{R}^n\), connectedness, path‑connectedness), Multivariable Calculus (chain rule, the differential of a smooth map).

Curriculum

Finite groups representations: core theory

1. Definition of a representation. \( G \)-modules.
2. Invariant subspaces, irreducible representations.
3. Morphisms of representations. Schur’s lemma.
4. Complete reducibility, averaging, Maschke’s theorem.
5. Uniqueness of the decomposition of a completely reducible representation into a sum of irreducible ones.

Constructions and characters

6. Tensor product of representations. Exterior and symmetric powers of representations. Complexification.
7. Examples of irreducible complex representations. Groups of isometries of the regular polyhedra.
8. Characters of representations. Group algebra. Regular representation.
9. Orthogonality relations of irreducible characters. Fourier transform of functions on a group; Plancherel formula.
10. Examples of complex character tables: abelian groups, the dihedral group \( D_n \), symmetric groups \( S_3 \), \( S_4 \), the alternating group \( A_4 \). Decomposition of tensor products of irreducible representations.
11. Burnside’s isomorphism: the group algebra of a finite group as a product of matrix algebras corresponding to irreducible representations.
12. * Outlook: Young tableaux and irreducible complex representations of \( S_n \).

Lie groups and Lie algebras: foundations

13. Matrix Lie groups. Connectedness. Compactness.
14. Orthogonal and symplectic groups. Heisenberg group.
15. Covering of \(\mathrm{SO}(3, \mathbb{R})\) by \(\mathrm{SU}(2)\).
16. The Lie algebra as the tangent space of a Lie group at the identity element. The exponential map. Abstract Lie algebras.

Representations of Lie algebras

17. The connection between representations of a Lie group and its Lie algebra. The case of connected or simply connected Lie groups.
18. The adjoint representations of a Lie group and its Lie algebra.
19. Representations of the Lie algebra \(\mathfrak{sl}(2, \mathbb{C})\). Polynomial representations of the Lie group \(\mathrm{SL}(2, \mathbb{C})\).
20. Tensor products of representations of a Lie algebra. Clebsch‑Gordan coefficients for the Lie algebra \(\mathfrak{sl}(2, \mathbb{C})\).
21. Compact Lie groups. Peter‑Weyl theorem.
22. * Cartan subalgebra, roots, and weights for \(\mathfrak{sl}(3, \mathbb{C})\); construction and classification of finite‑dimensional irreducible representations via highest weight.

* Topics marked with an asterisk will be covered if time permits.

Assessment and grading

The course meets twice per week: one lecture and one problem-solving session. All materials are shared in advance as pdf files and via a collaborative whiteboard (Miro).

Each week, you are expected to submit solutions to at least three problems across the boards. Each correct solution earns 1 point, up to 3 points per week. Additional bonus points may be awarded for particularly elegant or especially challenging solutions.

Both the midterm and final exams are oral, open-book, and open-notes. The midterm consists of five problems (4 points each), and the final consists of five problems (5 points each). You may also earn bonus points by identifying and correcting errors during lectures.

Final grade = (regular points + bonus points) / (maximum regular points).
Scores above 100% are possible. As a guideline: A+ ≥ 90%, A ≥ 85%, with lower thresholds assigned accordingly.

Textbooks

We recommend several textbooks below. Please select those that best meet your needs. You are not required to use the most recent editions; these are listed for your reference only.

• G. James, M. Liebeck, Representations and Characters of Groups, 2nd edition, 2001 (for items 1-11 (finite groups) of the curriculum).
• B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd edition, 2015 (for items 13-22 (Lie groups/algebras) of the curriculum).
• W. Fulton, J. Harris, Representation Theory: A First Course (Graduate Texts in Mathematics, vol. 129), corrected edition, 2004.
• E. B. Vinberg, Linear Representations of Groups, 2nd reprinting, 2010.

Teasers

1. Consider a die. At each step, replace the number on each face with the sum of the numbers on the neighboring faces. Find the six numbers on the faces of the cube after \(100\) steps.
2. Irreducible complex characters of a finite group \(G\) form an orthonormal basis for the space of central functions on \(G\).
3. For every natural number \(n\), there exists a unique representation (up to isomorphism) of the Lie algebra \(\mathfrak{sl}_2(\mathbb{C})\) of dimension \(n\).
4. The dimension of the subspace of fixed vectors in the \(n\)-th tensor power of the standard representation of the Lie group \(\mathrm{SL}_2(\mathbb{C})\) is equal to the \(n\)-th Catalan number.