Representation Theory

Brief description

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

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

Lecture VII: 3rd October

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

1. Definition of Symmetric Functions
2. Monomial, elementary and complete symmetric functions; transition matrices
3. Power sum symmetric functions and the character table of S_n

1. Twisting by characters
2. Intertwining twistsed permutation representations
3. Transpose partition
4. Twisting V_i by the sign character

1. Combinatorial Resolution Theorem
2. RSK correspondence
3. Decomposition of partition representations and classification of irreducible representations of S_n in the semisimple case.

1. Permutation Representations
2. Relative positions and intertwiners between permutation representations
3. Decomposition of subset representations
4. Decomposition of partition representations of S₃

1. Endomorphisms of completely reducible modules
2. Wedderburn decomposition
3. Primitive central idempotents in the group algebra

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