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. Examples of representations. Subrepresentations. Invariant subspaces. Irreducible representations.
  2. Direct sum of representations. Complete reducibility. Maschke’s theorem.
  3. Tensor product of representations.
  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, their properties. Conjugacy classes. Group algebra. Central functions. Regular representation.
  6. Orthogonality of complex matrix elements. Group algebra of a finite group as a product of matrix algebras, corresponding to irreducible representations.
  7. Number of irreducible complex characters. Orthogonality relations.
  8. Examples of complex character tables: abelian groups, dihedral group Dn, groups S3, S4, A4. Decomposition of tensor products of irreducible representations.
  9. Examples of Lie groups. Definition of a matrix Lie group. Connected Lie groups.
  10. Covering of SO(3,R) by SU(2).
  11. Lie algebra is the tangent space of a Lie group at the identity element. Definition of an abstract Lie algebra.
  12. Representations of the Lie algebra sl(2,C).
  13. Connection between representations of Lie groups and Lie algebras. Polynomial representation of the Lie group sl(2,C).
  14. Tensor product of representations of Lie algebras. Clebsch-Gordan decomposition for the Lie algebra sl(2,C).
  15. Compact groups and their representations. Peter-Weyl theorem.
  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

 

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