Course for summer interns in Representation Theory
01--15 June 2012 at the Institute of Mathematical Sciences (IMSc)

This course is aimed at IMSc summer 2012 interns in mathematics, but everybody is welcome. Please send an email message to knr at imsc dot res dot in in case you plan to attend or just want to be on the mailing list for the course.

Schedule: The first lecture will be on Monday 4th June at 1000 hrs in room 117. Subsequent lectures will be in Alladi Ramakrishnan Hall (room 423).

Syllabus: The basics will be covered of the linear representation theory over the complex numbers of finite groups: Schur's lemma, complete reducibility (Mashke's theorem), character theory, isotypical components and projection formulae, matrix coefficients, Fourier transform, induction and reciprocity. The concepts and results covered will be illustrated by the study of the case of symmetric groups.

Pre-requisites: We will assume knowledge of some basic linear algebra and abstract algebra (as taught in first and second year undergraduate mathematics courses). Participants are required to prepare themselves by learning some basic multi-linear algebra as for example featured in this video.

Speakers: Amritanshu Prasad, K. N. Raghavan, and S. Viswanath

Schedule: The morning sessions (0900--1300 hrs) of the week days will be occupied by lectures. There maybe some tutorial sessions in the afternoons depending upon demand. The participants will likely require to put in significant effort by themselves (several hours for every lecture hour) to be able to keep up with the course. They are encouraged to learn as much from discussions with one another as from the lectures or from books.

A rough plan of the lectures (click on title for notes if linked):

  1. G-sets: G-sets; orbits and stabilisers; class equation; applications to finite groups: Sylow theory.
  2. Basics: representations, invariant inner product (over complex numbers), diagonalizability, invariant subspaces, irreducibility.
  3. Complete reducibility: new representations from old ones, intertwiners, Schur's lemma, isomorphism, first projection formula, complete reducibility.
  4. Character theory: orthonormality of characters, second projection formula, isotypical components.
  5. Matrix coefficients: group algebra, Fourier transform.
  6. Induction: induced representations, restrictions.
  7. Induction (continued): Frobenius reciprocity, Mackey's lemma.
  8. Representations of the symmetric group
  9. Representations of the symmetric group (continued)
  10. Representations of the symmetric group (continued)