Representation Theory

Aug-Dec 2011 (2:15-3:30pm, Mondays)

Brief description

The goal of this course is to understand the representaion theory of symmetric groups:
  1. Characters of representations of finite groups in the semisimple case
  2. Permutation representations and their intertwiners
  3. Grassmannian representations of S_n and GL_n(F_q)
  4. RSK correspondence
  5. Classification of irreps of S_n (using 3 and 4)
  6. Dual RSK correspondence
  7. Tensoring by sign character (using 6)
  8. Symmetric polynomials
  9. Characteristic function
  10. Frobenius character formula
All lectures are recorded and can be accessed online from the lecture web page below or from the youtube playlist.

Recommended reading

  1. Symmetric functions and Hall polynomials, by I. G. MacDonald, Oxford University Press, 1979.
  2. Permutations, matrices, and generalized Young tableaux, by D. E. Knuth, Pacific J. Math. 34 1970, 709-727.

Topics covered

Click on the date for lecture notes, assignments and videos.

Lecture VII: 3rd October

  1. Schur functions: Kostka and Cauchy
  2. Frobenius Character Formula

Lecture VI: 19th September
  1. Definition of Symmetric Functions
  2. Monomial, elementary and complete symmetric functions; transition matrices
  3. Power sum symmetric functions and the character table of Sn

Lecture V: 12th September
  1. Twisting by characters
  2. Intertwining twistsed permutation representations
  3. Transpose partition
  4. Twisting Vi by the sign character

Lecture IV: 5th September
  1. Combinatorial Resolution Theorem
  2. RSK correspondence
  3. Decomposition of partition representations and classification of irreducible representations of Sn in the semisimple case.

Lecture III: 29th August
  1. Permutation Representations
  2. Relative positions and intertwiners between permutation representations
  3. Decomposition of subset representations
  4. Decomposition of partition representations of S3

Lecture II: 22nd August
  1. Endomorphisms of completely reducible modules
  2. Wedderburn decomposition
  3. Primitive central idempotents in the group algebra

Lecture I: 8th August:
  1. Representations and modules (the definitions)
  2. The group algebra, and how modules for the group algebra correspond to representations
  3. Invariant subspaces
  4. Simplicity and Schur’s lemma
  5. Projection Yoga
  6. Maschke’s theorem