Representation Theory
AugDec 2011 (2:153:30pm, Mondays)
Brief description
The goal of this course is to understand the representaion theory of symmetric groups:
 Characters of representations of finite groups in the semisimple case
 Permutation representations and their intertwiners
 Grassmannian representations of S_n and GL_n(F_q)
 RSK correspondence
 Classification of irreps of S_n (using 3 and 4)
 Dual RSK correspondence
 Tensoring by sign character (using 6)
 Symmetric polynomials
 Characteristic function
 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

Symmetric functions and Hall polynomials, by I. G. MacDonald, Oxford University Press, 1979.

Permutations, matrices, and generalized Young tableaux, by D. E. Knuth, Pacific J. Math. 34 1970, 709727.
Topics covered
Click on the date for lecture notes, assignments and videos.
Lecture VII: 3rd October
 Schur functions: Kostka and Cauchy
 Frobenius Character Formula
Lecture VI: 19th September
 Definition of Symmetric Functions
 Monomial, elementary and complete symmetric functions; transition
matrices
 Power sum symmetric functions and the character table of
S_{n}
Lecture V: 12th September
 Twisting by characters
 Intertwining twistsed permutation representations
 Transpose partition
 Twisting V_{i} by the sign character
Lecture IV: 5th September
 Combinatorial Resolution Theorem
 RSK correspondence
 Decomposition of partition representations and classification of irreducible representations of S_{n} in the semisimple case.
Lecture III: 29th August
 Permutation Representations
 Relative positions and intertwiners between permutation representations
 Decomposition of subset representations
 Decomposition of partition representations of S_{3}
Lecture II: 22nd August
 Endomorphisms of completely reducible modules
 Wedderburn decomposition
 Primitive central idempotents in the group algebra
Lecture I: 8th August:
 Representations and modules (the definitions)
 The group algebra, and how modules for the group algebra correspond to representations
 Invariant subspaces
 Simplicity and Schur’s lemma
 Projection Yoga
 Maschke’s theorem