MT4144 Representation Theory of Finite Groups
(August - November 22, IISER Pune)
Instructor : Dr Anupam Singh
Audience : 4th year BS-MS, iPhD, PhD students at IISER Pune
Schedule : Monday 10AM, Tuesday 2PM, Wednesday 12PM, LHC 108
Evaluation : Test I - 20%, mid-sem - 20%, Test II - 20%, end-sem - 40%
Prerequisite : Basic Algebra courses (Groups, Vector spaces, Modules etc)
Goal of the course : This course introduces the representation theory of finite groups via character theory. The subject of representation theory is quite deep and needed in advanced mathematics (such as number theory, Lie theory, Harmonic Analysis etc) courses as we all as Physics and Chemistry.
Proposed Content : Basic definitions and properties of representations, character theory and decomposition of regular representations, Maschke's Theorem, examples of irreducible representations and character tables, group rings, induced representations and Mackey theory. Burnside's pq Theorem.
Text books and reference material :
Representation Theory of Finite Groups (course notes): Anupam Singh
Representation Theory of Finite Groups: B. Steinberg
Linear Representations of Finite Groups : J. P. Serre
Representations and Characters of Groups : James and Liebeck
Representations of Finite and Compact Groups : B. Simon
Weekly topics covered :
08/08/22 Representation, invariant subspace, irreducible representation
09/08/22 Holiday
10/08/22 Examples/Tutorial
15/08/22 Holiday - extra class - sub representation, Direct sum, Maschke's Theorem, regular representation
16/08/22 Proof of Maschke's Theorem
17/08/22 Tutorial
22/08/22 Schur's Lemma
23/08/22 Representation of Abelian groups over C
24/08/22 Tutorial
29/08/22 Group Algebra, simple and semisimple module and rings
30/08/22 Matrix elements
31/08/22 Holiday
05/09/22 Character of a representation, basic properties
06/09/22 Summary of Character Theory and making character table
07/09/22 Tutorial
10/09/22 - Test 1 - at 2 PM in the Madhava Hall
12/09/22 Orthogonality of characters
13/09/22 Number of irreducible characters vs number of conjugacy classes
14/09/22 Tutorial
19/09/22 Character Table of groups up to order 9
20/09/22 Character table of A_4 and S_4
21/09/22 Character table of dihedral group
Mid-sem on 27 at 3PM
24-30 September - Mid sem exam week
2-5 October - Mid sem break
10/10/22 Constructing representations- subrepresentation, direct sum, restriction, tensor product (induced to be done later)
11/10/22 Symmetric power and exterior power
12/10/22 Character table of S_5
17/10/22 Central characters, irreducible degree divides order of G
18/10/22 Burnside's pq Theorem (see the original proof by Burnside)
19/10/22 Tutorial
24/10/22 Holiday
25/10/22 Induced representation, examples
26/10/22 Character of induced representation
Test 2 -- mini projects topics
Groups of order p^3 and character table
Characters of groups of order pq
Linear groups and certain subgroups
Real representations
Artin-Wedderburn decomposition
Brauer's theorem about the number of irreducible rep in char p
Probability and Random walks on groups
Graphs and Characters theory
Group actions and permutation representation
31/10/22 Induced representation via tensor product
01/11/22 No class
02/11/22 Frobenius Reciprocity
07/11/22 Mackey-Wigner irreducibility criteria
08/11/22 Holiday --- presentations of mini projects whole day
09/11/22 Presentation of mini projects
14/11/22 Characters of GL(2,q)
15/11/22 Presentation of mini projects
16/11/22 Presentations of mini projects
End sem exam 28/11/22