irreducible and indecomposable representations, Schur's Lemma and the Krull-Schmidt Theorem
induced representations
semi-simplicity and Wedderburn's Theorem
Representations of finite groups
Maschke's Theorem
Frobenius reciprocity
characters and idempotents
symmetric group and combinatorics of Young diagrams
Further topics (as time permits)
Schur-Weyl duality
Quiver representations
Homological algebra
Morita equivalence
Examination
Examination consists of weekly homework that you hand in electronically on Canvas (upload scanned JPG of hand written work, or LaTeX:ed PDF, or exported from iPad/tablet etc., all are OK if legible).
(due Feb 14) Homework 3 is to do Problem 2.15.1 in [Etingof], parts (a)–(g), and (m). If you already know Lie algebras, do [Problem 2.16.5] following similar steps instead, including finding a q-analog of the Casimir element.
(due Feb 21) Homework 4: Etingof Exercise 3.6.1, Problem 3.9.1 parts (a),(b),(c), and Problem 3.9.5
(due Feb 28) Homework 5: Etingof Problems 3.9.3 (you can assume the quiver is just two vertices and a single arrow), 4.12.1 and 4.12.2.
Lecture summaries and suggested reading
Below we abbreviate:
[L] = Lorenz, A Tour of Representation Theory
[E] = Etingof et al., Introduction to Representation Theory
Lecture 1 (Wed Jan 22): Covered first 8 pages of Lecture 1 Tensor products of vector spaces. Associative algebras. Read [E, §2.11, §2.2], [L, §1.1.1, §B.1.1, §B.3.1].
Lecture 2 (Fri Jan 23): Covered rest of Lecture 1 and part of Lecture 2 on Tensor products of algebras, examples of algebras, tensor/symmetric/exterior algebras, graded algebras. Read [E, §2.1, §2.2, §2.11.3, §2.12], [L, §1.1.2]
Lecture 3 (Mon Jan 27): [E, §2.3], representations and modules, intertwining operators, Schur's Lemma (any field)
Lecture 4 (Wed Jan 29): [E, §2.3–§2.6]: Schur's Lemma (algebraically closed field), ideals, quotients, presentation by generators and relations
Lecture 5 (Fri Jan 31): Basis of the Weyl algebra [E, §2.7], Lie algebras and enveloping algebra [E, §2.9, Def. 2.9.9, Def. 2.12.3], [L, §5.1.1, §5.4.1].
Lecture 6 (Mon Feb 3): Lie algebras and their representations; universal enveloping algebra [E, §2.9], [L, §5.1, §5.4]
Lecture 7 (Wed Feb 5): Quivers and their representations; path algebras [E, §2.8]
Lecture 8 (Fri Feb 7): Subrepresentations of semisimple representations [E, §3.1], [L. §1.4]
Lecture 9 (Mon Feb 10): Jordan Density Theorem, Representations of direct products of matrix algebras, [E, §3.1–3.3]
Lecture 10 (Wed Feb 12): Proof of Thm 3.3.1, Filtrations (composition series) [E §3.4], taking the quotient by the radical [E §3.5]
Lecture 11 (Fri Feb 14): Characterization of semisimple algebras [E, §3.5]. Characters of representations [E, §3.6]
Lecture 12 (Mon Feb 17): Jordan-Hölder and Krull-Schmidt Theorems [E, §3.2]
Lecture 13 (Wed Feb 19): Representations of Finite Groups: Maschke’s Theorem [E, §4.1], irreducible characters and class functions [E, §4.2]
Lecture 14 (Fri Feb 21): Quaternion group. Pulling back representations from quotients. The row orthogonality relation in character tables.
Old lecture notes from 2021
Below are old lecture notes from Spring 2021 (then it ran on Tue/Thu so each lecture was 75 minutes). We might move around the topics a bit and/or update the 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 highest weight representations.
Lecture 28 Center of universal enveloping algebra of the general linear Lie algebra. Capelli determinant. Branching rule. GZ basis.
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