NCM Workshop on Probability and Representation Theory

CONTENTS

Update: The list of selected candidates is here.

Update #2: Videos of some of the lectures are now available here.

Topics, Speakers and References

1. Representation Theory and Vershik-Okounkov Theory — Murali Srinivasan (IIT Bombay)
   All three books below are authored by Ceccherini-Silberstein, Scarabotti and Tolli.
   1. Harmonic analysis on finite groups, Cambridge University Press, 2008.
      This discusses the random transpositions model in detail (with background on Markov chains and representation theory included)
   2. Representation theory of the symmetric groups, Cambridge University Press, 2010.
      This is a book length discussion of the Okounkov-Vershik theory.
   3. Representation theory and Harmonic analysis of Wreath products of finite groups, Cambridge University Press, 2014.
   See also my lecture notes on Okounkov-Vershik theory.


2. Markov Chains — Antar Bandyopadhyay (ISI Delhi)
   
   1. For basic Markov chain theory, see this course webpage
   2. For random walks on finite groups, see this monograph by Laurent Saloff-Coste.


3. Computer Algebra — Amritanshu Prasad (IMSc Chennai)
     Explore the Sage documentation starting here.


4. Semigroup theory — Pooja Singla (IISc Bangalore)
     The reference for left-regular bands is this paper by K. Brown


5. Random walk models — Arvind Ayyer (IISc Bangalore)
   The details of the models are given below.

Research talks

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

Syllabus

Since the pioneering work of Diaconis and Shahshahani on generating random permutations using transpositions in 1981, representation theory has become an important tool in understanding random walks on finite groups. We will study certain natural random walk models on the symmetric group in increasing order of difficulty. In these models, one prescribes a probabilistic rule that describes what the possible permutations are at time n+1 if one has a certain permutation at time n (time is taken to be discrete). For each model, we will understand how a random permutation looks like after a long time and how long it takes for the distribution to reach stationarity. The first three models are well-studied; we will develop our understanding of the theory by working through them. In the end, we will discuss the fourth model, where some of the same questions remain open. Very recently this preprint has been posted to the arXiv announcing proofs of the eigenvalue conjectures in the latter model.

1. Random Transpositions – This is the original model of Diaconis and Shahshahani, where one obtains a new permutation at each step by performing a random transposition. The study of this model requires a basic understanding of Markov chains and character theory of the symmetric group.
   Reference: Generating a Random Permutation with Random Transpositions, P. Diaconis & M. Shahshahani, Z. Wahr. verw. Gebiete, 57(2):159-179, 1981.


2. Transpose-top-to-random shuffle – In this model, one obtains a new permutation at each step by making the transposition (1,j) where j is chosen at random. That is, the first symbol is interchanged with a uniformly randomly chosen symbol. The study of this model requires an understanding of the so-called Jucys-Murphy elements and the more recent Vershik-Okounkov theory of the representation of the symmetric group.
   Reference: Applications of Nonommutative Fourier Analysis to Probability Problems P. Diaconis, et al. Ecolé d' Été de Probabilites de St. Flours, XV-XVII, Springer Lecture Notes in Mathematics, 1362:51-100. Springer-Verlag, Berlin.


3. Tsetlin library – This is a simplistic model of a library with a single shelf! One obtains a new permutation at each step by multiplying the existing permutation with the cycle (1, 2, ... ,j) where again j is chosen at random. This model is qualitatively different from the previous two because it requires the study of the representation theory of semigroups. This is the simplest such model and the ideas involved in understanding this model follow naturally from ideas in the theory of group representations.
   Reference: A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements, P. Bidigare, P. Hanlon, D. Rockmore, Duke Mathematical Journal, Volume 99, Number 1, 135-174, 1999.


4. Random-to-random shuffle – This model is a natural generalisation of the top-to-random shuffle, but there are several conjectures about this model that remain wide open. One chooses two symbols i and j uniformly at random, and then multiplies the existing permutation by the cycle (i, i+1, ... , j-1, j) if i<=j or (j, j+1, ... , i-1, i) if i>j. One of the main aims of this workshop will be to explore this model, understand recent progress, and make some headway towards the open questions.
   Reference: Spectral analysis of random-to-random Markov chains, A. B. Dieker, Franco Saliola, arXiv:1509.08580.

Speakers and Schedule

Local organisers

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