Basic Representation Theory

Representation Theory studies how a group or other algebraic objects may act in vector spaces. It is a fundamental tool for studying groups using Linear Algebra. Representations appear in various models in Mathematical Physics, Number Theory, Algebraic Combinatorics, and other areas of mathematics. The course aims to introduce basic concepts and results of the classical theory of complex representations of finite groups and simple examples of representations of Lie groups and Lie algebras. In this course we consider only finite-dimensional representations.

Prerequisites: Linear Algebra, Elementary Finite Groups Theory.


Curriculum:

  1. Representations, G-modules. Invariant subspaces, irreducible representations. Rigid motions of the regular polyhedra.
  2. Complete reducibility, averaging, Maschke’s theorem.
  3. Tensor product of representations. Exterior and symmetric powers of representations. Complexification.
  4. Morphisms of representations. Schur’s lemma. Uniqueness of the decomposition of completely reducible representation into a sum of irreducible ones.
  5. Characters of representations. Group algebra. Regular representation.
  6. “Burnside’s isomorphism”: Group algebra of a finite group as a product of matrix algebras, corresponding to irreducible representations.
  7. Orthogonality relations. Fourier transform of functions on a group, Plancherel formula.
  8. Complex character tables examples: abelian groups, the dihedral group Dn, symmetric groups S3, S4, the alternating group A4. Decomposition of tensor products of irreducible representations.
  9. Matrix Lie groups. Connectedness. Orthogonal and symplectic groups. Heisenberg group.
  10. Covering of SO(3,R) by SU(2).
  11. Lie algebra is the tangent space of a Lie group at the identity element, exponential map. Abstract Lie algebra.
  12. Representations of the Lie algebra sl(2,C).
  13. Connection between representations of a Lie group and its Lie algebra. Adjoint representation. Polynomial representation of the Lie group SL(2,C).
  14. Tensor product of representations of a Lie algebra. Clebsch-Gordan coefficients for the Lie algebra sl(2,C).
  15. Compact groups, Peter-Weyl theorem. Weyl’s unitary trick.
  16. Highest weight sl(3,C)-modules.

Textbooks

  • G.James, M.Liebeck, Representations and Characters of Groups, 2004.
  • W. Fulton, J. Harris, Representation theory. A first course., Berlin: Springer, 1991. (Grad.Texts Math., v. 129)
  • E.B. Vinberg, Linear Representations of Groups, 1989.
  • B. Hall, Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, 2003.

Teaser problems and results