- MF 2:15pm – 3:30pm Carver 204

- multilinear algebra
- algebras, homomorphisms, modules
- 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

- Maschke's Theorem
- Frobenius reciprocity
- characters and idempotents
- Clifford theory
- symmetric group and combinatorics of Young diagrams

- universal enveloping algebra
- highest weight theory
- Weyl's character formula
- crystals

- 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 8 Maschke's Theorem. Schur's Lemma. Wedderburn's Theorem.
- 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 26 Universal enveloping 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.

Each lecture one problem is assigned.

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