Advanced Instructional School (AIS)- Representation Theory

• Representation theory of groups is one of the most beautiful subjects in mathematics which finds applications to many other subjects including Physics and Chemistry, besides being of fundamental importance to numerous areas of mathematics. The aim of this summer school will be to offer a first course in representation theory of finite as well as compact groups. After developing the general theory, the school will try to emphasize representation theory of specific groups such as SU(2), SL(2,F_q), and the symmetric groups, which forms building blocks for representation theory of more complicated groups. There will be some lectures on applications of representation theory to Number theory, as well as combinatorics, and perhaps an example from the work of the Fields medalist Okounkov's work. There will be some lectures too to bring out the historical perspective of the subject.

  Proposed Syllabus and Speakers

1. Ashwani Bhandari: Basic Review of group theory with notions about the group acting on sets, the class equation, some uses, such as Sylow's theorem quickly done, conjugacy classes, symmetric group, the structure of their conjugacy classes, semi-simple rings and modules, Wedderburn structure theorem, application to the structure of group ring, and to representation theory of the group.
2.S. V. Kanetkar: Representation theory of finite groups with basic theorems proved without recourse to Wedderburn theory. Character table, Induced representations, Mackey irreducibility theorem, Clifford theory. Examples.
3. Maneesh Thakur: Representation theory of the symmetric group in terms of the Young tableau. The Schur-Weyl duality between GL(n) and S_m.
4. Ravi Kulkarni: Introduction to representation theory of compact Lie groups, with special emphasis on SU(2).
5. Amritanshu Prasad: Representation theory of GL(2,F_q), SL(2,F_q). Beginning of the story for GL(n,F_q) with parabolic induction, and irreducibility of it.
6. M .K. Srinivasan: Combinatorial applications of representation theory.

References:
1. J-P. Serre, Linear Representations of Finite Groups.
2. C. Musili, Representations Finite Groups, Hindustan Book Agency, 1993.
3. W. Fulton, J. Harris, Representation Theory: A first course, Graduate Texts in Mathematics.
   Readings in Mathematics 129. Springer Verlag (1991). International Edition (Low priced Ed.)
4. B. Simon: Representations of Finite and Compact Groups, Graduate Studies in Mathematics, vol. 10, AMS (1997).
5. E. B. Vinberg: Linear Representations of Groups, Birkhäuser (1988).

UM Lectures

1. I.B.S. PASSI : Historical Overview of Representation Theory

2. R. SRIDHARAN : Meromorphicity of Artin L functions

Note: All the participants will be given the book ``Representation Theory, A First Course", by W. Fulton, J. Harris (Springer).

Associate Teachers
1. Uma Ayer email: uma.iyer at bcc.cuny.edu
2. Anupam Kumar Singh email: anupam at math.tifr.res.in
3. Pooja Singla e-mail: pooja at imsc.res.in