Math 649: Representation theory.

This course is intended to become a regular offering in the ‘algebra and number theory’ collection of courses and so is not really a 649; rather it is the official ‘algebra and number theory’ offering this Spring. Here is a list of topics that are planned for this course.

Linear representations of finite groups: basic theory, group algebras, Schur’s lemma, Maschke’s theorem, the regular representation, character theory, induced representations, Frobenius reciprocity, examples, applications.

Representation theory of semisimple Lie algebras: the basics of Lie algebras, representations of $\mathrm{sl}_2(\mathbb C)$, representations of higher rank semisimple Lie algebras, highest weight theory, roots, relation to the representation theory of semisimple Lie groups.

Further topics may include (some but not all of): in-depth examples such as the symmetric group or finite linear groups, Burnside’s theorem, applications to combinatorics, harmonic analysis on finite groups, modular representation theory, representation theory of compact groups, the classification of complex semisimple Lie algebras and Dynkin diagrams.

Textbook: Fulton and Harris’ “Representation theory, a first course”, supplemented with other texts as necessary.

Instructor: Robert Harron