Representation Theory of Finite Groups

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

1. Etingof Pavel, Golberg Oleg, Hensel Sebastian, Liu Tiankai, Schwendner Alex, Vaintrob Dmitry, Yudovina Elena, Introduction to representation theory. With historical interludes by Slava Gerovitch, Student Mathematical Library 59. American Mathematical Society. 2011.
2. J. P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics. Vol. 42. Springer-Verlag. New York-Heidelberg. 1977.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.