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.
Lecture 22 Hook-length formula. Murnaghan-Nakayama rule. A word on profinite groups.
Lecture 23 Lie groups. One-parameter subgroups. Lie algebra of a matrix group. The bracket.
Lecture 24 Lie algebras and their maps, subalgebras, ideals, quotients, center, derived subalgebra. Functor from associative algebras. General linear Lie algebra and representations.
Lecture 25 Simple Lie algebras. Modules. Adjoint module. Special linear Lie algebra.
Lecture 27 Classical Lie algebras. Classification Thm for simple Lie algebras. Cartan subalgebras and root space decompositions. Weight and heighest weight representations.
Lecture 28 Center of universal enveloping algebra of the general linear Lie algebra. Capelli determinant. Branching rule. GZ basis.
Examination
Each lecture one problem is assigned.
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