AIS Representation Theory of Finite Groups (2017)

Venue: Chennai Mathematical Institute,  Chennai, Tamil Nadu
Date:  12th, Jun 2017 to 1st, Jul 2017

 


 

School Convener(s)

Name Amritanshu Prasad K. N. Raghavan K.V. Subrahmanyam
Mailing Address Institute of Mathematical Sciences
C. I. T. Campus,Taramani
Chennai 600 113
Institute of Mathematical Sciences
C. I. T. Campus,Taramani
Chennai 600 113
Chennai Mathematical Institute
SIPCOT IT PARK
Siruseri, Chennai - 600 103

 


 

Speakers and Syllabus 

 The entire contents of the book Representation Theory: A Combinatorial Viewpoint by Amritanshu Prasad, (Cambridge University Press, 2015) were covered in six modules. Chapter numbers in the following descriptions of these modules refer to Chapters in this book. The lecturers were (in chronological order of appearance) K N Raghavan (KNR), Amritanshu Prasad (AP) and S. Viswanath (SV), all from the Institute of Mathematical Sciences, Chennai. The teaching assistants were B. Ravinder (INSPIRE faculty fellow) and Shraddha Srivastava (senior research fellow), both from CMI

Module 1: Basics of linear representations of finite groups (KNR)
(Loosely following Chapter 1)

group actions: definitions and motivation through symmetry; linear representations: module theory of the group ring simple and semisimple modules; isotypic components; Schur's lemma; endomorphism ring of a semisimple module simple and semisimple rings; Wedderburn's structure theorem; averaging and Maschke's theorem  character theory projection formula (to any isotypic component); primitive central idempotents in group rings groups actions on sets: orbits, stabilizers, and invariant subsets; transitive actions; structure of an orbit; orbit counting lemma

Module 2: Young tableaux and the RSK correspondence (AP)
(Loosely following Chapter 3)

Semistandard Young tableaux and Kostka numbers.necessary and sufficient condition for the positivity of Kostka numbers The Robinson-Schensted-Knuth correspondence using Viennot's light-and-shadows method the dual RSK correspodence and its relationship to the Gale-Ryser theorem.

Module 3: Irreducible Representations of Symmetric Groups (AP)
(Loosely following Chapters 2, 4 and part of 5)

Permutation representations of symmetric groups and their intertwiners The classification of irreducible representations using permutation representations and the RSK correspondence  Twisting by sign character and dual RSK correspondence
Representation theory of alternating groups.

Module 4: Symmetric Functions-1 (SV)
(Loosely following parts of Chapter 5)

Symmetric functions: Monomial, elementary, complete homogeneous bases. Their transition matrices, using the RSK and dual RSK bijections. The power sum and Schur function bases, transition matrices in terms of characters of the permutation representations and irreducible representations of symmetric groups. The Inner product on the space of symmetric functions, pairs of dual bases, orthonormality of Schur functions.The classical definition of Schur functions as a quotient of determinants.

Module 5: Symmetric Functions-2 (SV)
(Loosely following parts of Chapter 5)

 Pieri rules, Murnaghan-Nakayama rule (and its recursive version).The Frobenius characteristic map, branching rule (from S_d to S_{d-1})Principal specialization of Schur functions, the hook-content formula for the number of semistandard tableaux, the hook length formula. Skew Schur functions, the Littlewood-Richardson rule.

Module 6: Polynomial representations of GL(n)
(Closely following chapter 6)
 polynomial representations of GL(n)The Schur algebras S(k) and S(n,k); homogeneous represenations structure and semisimplicity of the Schur algebra S(n,k); complete reducibility Schur-Weyl duality; modules of a commutant; classification of irreducibles
restrictions to the torus; characters  the character formula: the character as a Schur function

 


 


Time Table

Day

Date

Lecture 1
09:30–11:00

Lecture 2
11:30–13:00

Tutorial 1
14:15 to 15:15

Tutorial 2
15:45–16:45

1

12th

I

II

I

II

2

13th

I

II

I

II

3

14th

I

II

I

II

4

15th

I

II

I

II

5

16th

I

II

I

II

6

17th

I

II

I

II

7

19th

III

IV

III

IV

8

20th

III

IV

III

IV

9

21st

III

IV

III

IV

10

22nd

III

IV

III

IV

11

23rd

III

IV

III

IV

12

24th

III

IV

III

IV

13

26th

VI

V

V

VI

14

27th

V

VI

V

VI

15

28th

V

VI

V

VI

16

29th

V

VI

V

VI

17

30th

V

VI

V

VI

18

01st

V

VI

V

VI

 Tutorial sheets and notes for some of the lectures are available at  https://www.imsc.res.in/~knr/past/aisrt2017/index.html


 

 

Actual Participants 

Sr.

Full name

Gender

Affiliation

Position in College/ University

University/ Institute M.Sc./ M.A.

1

Mr. Swaraj Sridhar Pande

Male

Indian Statistical Institute, Bangalore Centre

B.Math Student

 

2

Mr. Vignesh S

Male

Indian Statistical Institute

B Math Student

 

3

Mr. Sajith Padinhatteeri

Male

Indian Institute of Technology Bombay

Research Associate

Pondicherry University

4

Mr. Mukesh Kumar Nagar

Male

IIT Bombay

PhD

Central University of Rajasthan

5

Mr. Kundan Singh

Male

IIT

MSc Student

IIT , KHARAGPUR

6

Mr. Hiranya Kishore Dey

Male

IIT BOMBAY

PhD

Ramkrishna Mission Vivekananda University

7

Mr. Basudev Pattanayak

Male

IISER Pune

PhD

IIT Bombay

8

Mr. N. Annamalai

Male

Bharathidasan University

PhD

Bharathidasan University/ M.Sc

9

Mr. Swaroop Hegde

Male

NISER Bhubaneswar.

M.Sc Student

 

10

Mr. Shantanu Sardar

Male

IIT KANPUR

PhD

JADAVPUR UNIVERSITY

11

Mr. Sourav Saha

Male

Indian Institute of Information Technology,Sri City

MS by Research

Indian Institute of Information Technology, Sri City

12

Mr. Sajal Kumar Mukherjee

Male

Visva-Bharati

PhD Student

Ramakrishna Mission Vivekanda University, Belur Math.

13

Mr. Sudip Bera

Male

Visva-Bharati

PhD

Ramakrishna Mission Vidyamandira, Belur Math. C. U.

14

Dr. Rameez Raja

Male

HRI

PDF

University of Kashmir

15

Dr. B Ravinder

Male

Tata Institute of Fundamental Research

Post-doctoral Fellow

University of Hyderabad

16

Ms Shraddha Srivastava

Female

Chennai Mathematical Institute

PhD

IIT Bombay, MSc

17

Mr. Shivang Jindal

Male

IIT Kanpur

B.S Student

 

18

Mr. Chinthalagiri Venkata Sriram

Male

Chennai Mathematical Institute

BSc Student

 

19

Mr. Himanshu Gupta

Male

Indian institute of Science, Bangalore

Integrated Ph.D student

Indian INstitute of Science,Bangalore

20

Mr. Subhajit Ghosh

Male

Indian Institute of Science

PhD

Indian Institute of Science

21

Ms Nimisha Pahuja

Female

Indian Institute of Science

Int PhD student

Indian Institute of Science

22

Mr. Surjadipta De Sarkar

Male

Indian Institute of Science

Int PhD student

Indian Institute of Science

23

Mr. Shib Proshad Pramanik

Male

Bhaskaracharya pratisthan

Rwsearcher

Tezpur university

24

Mr. Sai Praveen Madireddi

Male

Indian Institute of Science Education and Research - Mohali

BS-MS

 

25

Mr. Naveenkumar Sureshkumar Yadav

Male

BKM Science College

Asst. Prof

The Maharaja Sayajirao University of Baroda

 

26

Mr. Prashant Shivaji Arote

Male

TIFR,Mumbai

Ph.D. Student

Department of Mathematics,University of Pune.

27

Ms. B Prasuna

Female

Tata Institute of fundamenta research

PhD 1st year

University of Hyderabad

28

Ms Aparna Pradeep V.K.

Female

Cochin University of Science And Technology

Student

Cochin University of Science And Technology

29

Mr. Praneel Samanta

Male

UM-DAE CBS

Junior Project Assistant

UM-DAE CBS

30

Mr. Yogendra Sahu

Male

National Institute of Technology

M.Sc Student

M.Sc.

31

Mr. Sahil Singhal

Male

 

 

 

32

Mr. Sounak Paul

Male

Indian Statistical Institute, Bangalore Centre

B.Math Student

 

33

Mr. Kaurag Mukherjee

Male

Indian Statistical Institute, Bangalore Centre

B.Math (Hons.)

 

34

Mr. Prabhanka Deka

Male

Indian Statistical Institute, Bagalore

B.Math student

 

35

Ms. Raneeta Dutta

Female

IIT Madras

M.Sc Student

IIT Madras

36

Mr. Saikat Panja

Male

IISER Kolkata

Integrated phd 3rd year

IISER kolkata

37

Mr Saravanan V

Male

Periyar University

PhD

75

38

Mr. B Janaki Raman

Male

Indian Institute of Space Science and Technology

PhD student

RIASM, University of Madras

39

Mr. Giftson Santhosh.P

Male

Ramanujan Institute for Advanced Study in Mathematics

MSc student

Ramanujan Institute for Advanced Study in Mathematics

40

Mr. Rudra Narayan Padhan

Male

NIT, Rourkela

PhD

Indian institute of Technology Bombay

41

Mr. Nihar Prakash Gargava

Male

IIT Kanpur

Student

 

42

Mr. Harshit Yadav

Male

IIT, Kanpur

undergraduate student

NA

43

Mr. Sivasankar Chandrasekharan Nair

Male

Indian Institute of Technology, Kanpur

B.S. Student

Not applicable

44

Dr. Dilpreet Kaur

Female

IISER Pune

Post Doc Fellow

Panjab University

45

Mr. Ghosh Rahul

Male

ISI, Bangalore

B.Math.Hons

 

46

Mr Navnath Balasaheb Daundkar

Male

Chennai Mathematical Institute

PhD student

Pune university

47

Mr Shubham Sinha

Male

Indian Institute of Science

BSc (4th year)

 

 


How to reach

  • Airport: Chennai International Airport (code: MAA)
  • Chennai is well-connected by rail from various cities in India.
  • CMI is located in the Siruseri IT Park, off Old Mahabalipuram Road, connecting Chennai to Mahabalipuram. It is accessible by taxis from the airport and, additionally, buses from the city. Both the airport and the city are about 30km from the institute.
  • Getting to CMI