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Representation Theory of Finite Groups

Prof. R. Venkatesh · IISc Bangalore

Start with lecture 1 At NPTEL

Lectures

Week 1 Review: Basic Linear Algebra

  1. Basics of vector spaces Video
  2. Inner product spaces Video
  3. Diagonalization Video
  4. Generalized eigen spaces Video
  5. Decomposition theorems Video

Week 2 Representation Theory: An introduction through basic linear algebra, Representations of finite cyclic groups

  1. Definition of representation Video
  2. Isomorphism theorems Video
  3. Indecomposable representations Video
  4. Schur's Lemma Video
  5. Filtration of representations Video

Week 3 Representations of groups and algebras: definitions, examples, indecomposable (decomposable) representations,…

  1. Construction of new representations by extensions Video
  2. Basics of representation theory of finite groups Video
  3. Classification of irreducible representations of finite cyclic group and Symmteric group Video
  4. Irreducibility via Cyclic Vectors Video
  5. Construction of representations of group generated by a set with defining relations Video

Week 4 Complete reducibility of finite dimensional representations of finite groups and basic properties of character…

  1. Complete reducibility of unitary representations Video
  2. Complete reducibility of finite dimensional representations of finite groups Video
  3. Projection map from representation of G to its fixed subspace Video
  4. Character Theory: Basic properties Video
  5. Orthogonality relations of irreducible characters of finite group Video

Week 5 Schur orthogonality relations, regular representation, and its decomposition

  1. Orthonormal subset of space of class functions and Orthogonality of irreducible characters Video
  2. Schur's orthogonality relations and its applications Video
  3. Applications of character theory Video
  4. Regular representation, decomposition of regular representation Video
  5. Orthonormal basis of space of class functions and computation of character tables Video

Week 6 Second Orthogonality relations, Classification of irreducible characters of abelian group, Dimension theorem…

  1. Orthogonality of the columns of the character table Video
  2. Representations of finite abeilian group Video
  3. Algebraic numbers and equivalent conditions for algebraic integers Video
  4. Dimension theorem Video
  5. Applications of dimension theorem Video

Week 7 Character tables of some finite groups, Burnside Theorem, Tensor product space

  1. Character table computation of groups: Q_8, D_8, A_4 Video
  2. Burnside Theorem and its proof Video
  3. Proof of key result used in Burnside Theorem Video
  4. Notion of Tensor product Video
  5. Basic properties of Tensor product Video

Week 8 Tensor product, Symmetric tensor, alternating tensor product

  1. Symmetric tensor product Video
  2. Alternating tensor product Video
  3. G-action on Symmetric and exterior tensor product Video
  4. Graded character of Symmetric tensor algebra and exterior algebra Video
  5. Alternate proof of orthogonality of irreducible characters Video

Week 9 Density theorem, Frobenius theorem, Induced representation, Frobenius reciprocity and Mackey's restriction…

  1. Density theorem Video
  2. Frobenius theorem Video
  3. Induced representation Video
  4. Frobenius reciprocity Video
  5. Mackey's restrcition formula Video

Week 10 Mackey's irreducibility criterion, Fourier analysis on finite groups, Fourier transform, Cayley graph

  1. Mackey's irreducibility criterion Video
  2. Fourier analysis on finite groups Video
  3. Fourier inversion Video
  4. Fourier analysis on nonabeilian groups Video
  5. Applications of Fourier analysis on finite groups Video

Week 11 Eigenvalues of Cayley graph, combinatorics in symmetric group and introduction to representation theory of…

  1. Cayley graph and its eigenvalues Video
  2. Notations and combinatorics in representation theory of symmetric group Video
  3. Irreducible representations of symmetric group Video

Week 12 Permutation modules, construction and classification of irreducible representations of symmeric group

  1. Permutation modules Video
  2. Constructing irreducibles of symmetric group Video
  3. Classification of irreducible representations of symmetric group Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Teacher
Prof. R. Venkatesh
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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