MU-402 Representation Theory

Unit · year 4

MU-402 · Representation Theory

Threads structure24 lectures4 theorems

Groups realised as symmetries of vector spaces.

PREREQUISITES

L01	Why Represent a Group? —
L02	Representations and G-Modules —
L03	Subrepresentations and Irreducibility —
L04	Direct Sums and Complete Reducibility Maschke's theorem
L05	Maschke's Theorem Maschke's theorem
L06	Failure in Positive Characteristic Maschke's theorem
L07	Homomorphisms of Representations Schur's lemma
L08	Schur's Lemma Schur's lemma
L09	Consequences for Abelian Groups Schur's lemma
L10	The Group Algebra —
L11	Wedderburn's Structure Theorem —
L12	Characters: Definition and Basic Properties Orthogonality of characters
L13	Class Functions Orthogonality of characters
L14	Orthogonality of Characters Orthogonality of characters
L15	The Character Table Orthogonality of characters
L16	Constructing Character Tables Orthogonality of characters
L17	Decomposing Representations by Character Orthogonality of characters
L18	Induced and Restricted Representations —
L19	Frobenius Reciprocity —
L20	Permutation Representations Burnside's counting lemma
L21	Burnside's Counting Lemma Burnside's counting lemma
L22	Pólya Enumeration Burnside's counting lemma
L23	Representations of the Symmetric Group —
L24	Synthesis: Groups Seen Through Matrices Maschke's theorem, Orthogonality of characters

Representations of finite groups over C are completely reducible.

Morphisms between irreducibles are zero or isomorphisms.

Irreducible characters form an orthonormal set.

Orbits counted by average fixed points.