ATMW Probabability and Representation Theory (2016)

 There were five lecture courses in all. Listed below in separate subsections are the synopses of the courses in the words of the respective speakers.

2.1. Vershik-Okounkov theory for representations of the symmetric group: 4 lectures by Murali Srinivasan of IIT Mumbai. The first two lectures covered basic (ordinary) representation theory of finite groups. The third and fourth lectures covered the Vershik- Okounkov approach to the representation theory of the symmetric group. These results
were used to determine eigenvalues of the transitions matrices involved in the transpose- top-to-random and the random-transposition models of card shuffling.

2.2. Markov chains: 2 lectures by Antar Bandyopadhyay of ISI Delhi. Topics covered:Discrete Markov chains with countable state space, Examples including two state chain,Simple Symmetric Random Walk, random walk on d-dimensional integer lattice, Ehren-fest chain, and card shufflings. Classification of states, recurrence and transience; irre-ducibility, decomposition of state space into irreducible classes, examples. Stationary distributions, limit theorems, positive and null recurrence. Definition of periodicity, ape-riodic chains. Limit theorems for aperiodic irreducible chains, proof for positive recurrent case using coupling. Total variation distance and definition of mixing time.

2.3. Computer algebra: 2 lectures by Amritanshu Prasad of IMSc. Topics covered: In-troduction to Sage, basics of python programming, construction of some transition ma-trices using Sage. Irreducible representations of symmetric groups in Sage. Use of Sage’s
symmetrica interface to compute character values of symmetric groups.

2.4. Semigroup theory: 2 lectures by Pooja Singla of IISc. Topics covered: Interplay between representation theory of semigroup algebras and Markov chains on semigroups.Semilattice and Left regular band semigroups. Eigenvalues of Tsetlin library.

2.5. Card shuffling models: 4 lectures by Arvind Ayyer of IISc. The following random walk models on the symmetric group were discussed, with emphasis on the stationary distribution, eigenvalues of transition matrices, and mixing times:

(1) Random transpositions
(2) Transpose-top-to-random model
(3) Tsetlin library
(4) Top-to-random insertions
(5) Random-to-random insertions

There were four half-hour research talks as follows:

• Amritanshu Prasad (IMSc): Representations of the symmetric group with non-trivial determinant
• Sushil Bhunia (IISER, Pune): z-classes in unitary groups
• Ashish Mishra (CMI): Vershik-Okounkov approach to representations of the wreath product
• Hassain M (IISc): Representations of SL₂ (Z/16Z) and SL₂ (F₂ [t]/t⁴ )

Speakers and Schedule

Local organisers

Arvind Ayyer (IISc Bangalore)
Amritanshu Prasad (IMSc Chennai)
K N Raghavan (IMSc Chennai)

How to reach

The Chennai neé Madras, and to IMSc, the Institute of Mathematical Sciences. Here is some information on getting to the Institute. Address and phone numbers are

THE INSTITUTE OF MATHEMATICAL SCIENCES
C I T CAMPUS, TARAMANI, CHENNAI 600 113
Telephone Numbers

Here are some maps to give you a rough idea of the layout of Madras and the area around the Institute.

• A simplified schematic map of Madras, showing the major roads and Hotels and marking the area in which the CIT Campus is located.
• A postscript version of this figure if you want to download it to print at your end.
• A detailed Madras map.
• A gzipped postscript version of this figure
• A schenatic map of the location of IMSc within CIT Campus and its environs
• Detailed map of the location of IMSc within CIT Campus and other landmarks in the Greater Adyar area.

The Institute of Mathematical Sciences is located in a campus-like area variously known as the ``CIT Campus'' or the ``CPT Campus''. Many people in Madras may not have heard of the Institute, but almost everyone would have heard of the M.G.R. Film City and Tidel Park which are further down in the same campus area. IMSc has two complexes. One houses the main Institute building, the library building and the new building. The other has the students' hostel and Guest House. These two complexes are across the road from each other.