IST - Representation Theory (of finite groups) (2019)
Speakers and Syllabus
Module A (Viji Thomas, IISER TVM) Group action on sets: Permutation representations: the notion of action of a group: permutation, linear, and other types of representations of a group; relation to the notion of symmetry; examples; induced actions (using one or more given actions to produce new actions); orbits, stabilizers, invariant subsets; transitive actions and homogeneous spaces: structure of an orbit; lots of examples of computations of orders of orbits, centralizers; counting orbits: the Cauchy-Frobenius-Burnside lemma and applications, applications to group theory: e.g., to Sylow theorems.
Module B (R Venkatesh, IISc) Ordinary representation theory of finite groups via characters: basic results: Notion of a linear representation; examples. Notion of a group ring: why representations are the same as modules over the group ring; invariant subspaces and irreducible representations; Schur’s lemma (various versions); semisimplicity of ordinary representations of a finite group using the idea of averaging; notion of character; irreducible characters form an orthonormal basis for the Hilbert space of class functions; calculations using characters; computation of character tables; matrix coefficients
and their orthonormality.
Module C (Geetha T, IISER TVM) Basics of the Wedderburn structure theorem for a finite dimensional semisimple algebra: application to ordinary representation theory of a finite group: Simple and semisimple modules; simple and semisimple rings; Matrix rings as examples of simple finite dimensional algebras; semisimplicity of the group ring (over a field of char. zero) of a finite group; Wedderburn structure theorem for a semisimple finite dimensional algebra over an algebraically closed field; applications to ordinary representation theory of a finite group; structure of a simple ring: Burnside’s lemma (about the map to the endomorphism ring of a simple finite dimensional module being over an algebraically closed field being onto) Module D (K N Raghavan, IMSc) Examples: especially the case of symmetric groups. Tabloid representations and their decomposition. Specht modules. Young tableaux and RSK correspondence.
Module D (K N Raghavan, IMSc) Examples: especially the case of symmetric groups. Tabloid representations and their decomposition. Specht modules. Young tableaux and RSK correspondence.
1. K N Raghavan (KNR), IMSc
2. Geetha Thangavelu (GT), IISER TVM
3. Viji Thomas (VT), IISER TVM
4. R Venkatesh (RV), IISc
Course Associates (tentative)
1. Bhimarthi Ravinder (BR), INSPIRE faculty, CMI
2. Shraddha Srivastava (SS), PDF, IMSc
|9.30 - 11.00
|June 03||VT||RV||VT, BR, SS||RV, BR, SS|
|June 04||VT||RV||VT, BR, SS||RV, BR, SS|
|June 05||VT||RV||VT, BR, SS||RV, BR, SS|
|June 06||VT||RV||VT, BR, SS||RV, BR, SS|
|June 07||VT||RV||VT, BR, SS||RV, BR, SS|
|June 08||VT||RV||VT, BR, SS||RV, BR, SS|
|June 10||GT||KNR||GT, BR, SS||KNR, BR, SS|
|June 11||GT||KNR||GT, BR, SS||KNR, BR, SS|
|June 12||GT||KNR||GT, BR, SS||KNR, BR, SS|
|June 13||GT||KNR||GT, BR, SS||KNR, BR, SS|
|June 14||GT||KNR||GT, BR, SS||KNR, BR, SS|
|June 15||GT||KNR||GT, BR, SS||KNR, BR, SS|
Tea Breaks: 11.00 - 11.30 and 3.30 - 4.00
Lunch Break: 1.00 - 2.30
Snacks: 5.00 - 5.30