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.
Each lecture one problem is assigned.
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