irreducible and indecomposable representations, Schur's Lemma and the Krull-Schmidt Theorem
induced and coinduced representations, adjointness relations
primitive ideals, Jacobson radical
semisimplicity and Wedderburn's Theorem
Representations of finite groups (Part II)
Maschke's Theorem
Frobenius reciprocity
characters and idempotents
Clifford theory
symmetric group and combinatorics of Young diagrams
Highlights of representation theory for Lie algebras (Part III)
universal enveloping algebra
highest weight theory
Weyl's character formula
crystals
Lecture notes
Lecture 1 Tensor products of vector spaces. Associative algebras.
Lecture 2 Tensor, symmetric, and exterior algebras.
Lecture 3 Determinant and top exterior power. Modules §1.1.3. Restriction functor §1.2.2.
Lecture 4 Tensor-Hom adjunction. Modules vs Representations. Group algebra and group representations.
Lecture 5 Subreps. Irreducible and indecomposable reps. Sign rep of symmetric group. Natural rep of dihedral group. First look at induced representations.
Lecture 6 Tensor products over noncommutative algebras. Extension of scalars. Induced modules. Standard representation of symmetric group.
Lecture 7 General tensor-hom adjunction for bimodules. Two realizations of the restriction functor. Application: Adjoint sequence Ind⊣Res⊣Coind.
Lecture 9 Details on Wedderburn's Thm. Consequences for group representations. Commutator trick and counting irreps via conjugacy classes.
Lecture 10 Characters and class functions. Additivity. Character tables. Cyclic groups. Orthogonality relation.
Lecture 11 Duals, tensor products, and Homs between reps. Symmetrizing idempotent and dimension of space of invariants. Proof of (row) orthogonality relation. Characters characterize!
Lecture 12 Self-duality. Tensoring with one-dimensional reps. The representation ring of a group. Inflation.
Lecture 13 Character table of symmetric group on 4 elements. Column orthogonality relation. Character of the standard representation of the symmetric group. Irreducibility criterion for characters. Character of the wedge square and symmetric square of a representation.
Lecture 14 Character table for symmetric group on 5 elements and its alternating subgroup. Clifford's Theorem (index 2 case).
Lecture 15 Automorphism-twisted representations. Action of G on Rep N, for a normal subgroup N of G. Clifford's Theorem (general case).
Lecture 16 Integral closure. Character values are algebraic integers. Irreps produce central characters.
Lecture 17 Frobenius' and Schur's divisibility theorems.
Lecture 18 First lecture on Okounkov-Vershik approach to rep theory for symmetric group. Gelfand-Zetlin subalgebra. Jucys-Murphy elements. Olshanskiĭ's centralizer theorem.
Lecture 19 The JM elements generate the GZ subalgebra. Multiplicity-free branching rule. The branching graph. GZ bases of irreps via paths.
Lecture 20 GZ basis of the standard rep of the symmetric group. Thm: GZ subalgebra acts diagonally in GZ basis, is maximal commutative and semisimple. Dimension of GZ subalgebra.
Lecture 21 Joint spectrum of JM elements. Partitions, Young diagrams and the Young graph. Paths as standard Young tableaux (SYTs). Contents of a SYT. Young graph is isomorphic to the branching graph.
Examination
Each lecture one problem is assigned.
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