Representation Theory of Finite Groups

ABOUT THE COURSE:Representation theory is a branch of mathematics dedicated to exploring abstract algebraic structures by representing their elements as linear transformations of vector spaces. This field has numerous fascinating applications in both mathematics and physics. In this course, we will focus on the representation theory of finite groups.PREREQUISITES: First course in linear algebra and basic algebra

Syllabus

Week 1: Review of Linear algebra: Inner product spaces, diagonalizations.
Week 2:Representation theory: an introduction through basic linear algebra. Representations of finite cyclic groups.
Week 3:Representations of algebras: definitions, examples, indecomposable/decomposable representations, reducible/irreducible representations.
Week 4:Representation of finite groups: Maschke’s Theorem on complete reducibility, Schur’s lemma.
Week 5:Character theory: orthogonality of irreducible characters, class functions
Week 6:Induced characters and Frobenius reciprocity
Week 7:The regular representation. Decomposition of regular representation. Number of irreducible representations.
Week 8:The complex function space on G, orthogonality of co-efficient functions.
Week 9:Tensor product of vector spaces, tensor product of representations, sym and alt constructions
Week 10:Dimension theorem, Burnside’s theorem. Induced Representations and Frobenius reciprocity
Week 11:Mackey’s Irreducibility Criterion
Week 12:Representation theory of the symmetric group: the Specht modules of partitions of n