# 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 (Sandeep Varma, TIFR Mumbai) 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.

Speakers
1. K N Raghavan (KNR), IMSc
2. Geetha Thangavelu (GT), IISER TVM
3. Viji Thomas (VT), IISER TVM
4. Sandeep Varma (SV), TIFR Mumbai

Course Associates (tentative)
1. Bhimarthi Ravinder (BR), INSPIRE faculty, CMI
2. Shraddha Srivastava (SS), PDF, IMSc

## Time Table

 9.30 - 11.00Lecture 1 11.30-1.00Lecture 2 2.30-3.30Tutorial 1 4.00-5.00Tutorial 2 June 03 VT SV VT, BR, SS SV, BR, SS June 04 VT SV VT, BR, SS SV, BR, SS June 05 VT SV VT, BR, SS SV, BR, SS June 06 VT SV VT, BR, SS SV, BR, SS June 07 VT SV VT, BR, SS SV, BR, SS June 08 VT SV VT, BR, SS SV, 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

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