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