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Since the pioneering work of Diaconis and Shahshahani on generating random permutations using transpositions, representation theory has become an important tool in understanding random walks on groups.
The aim of this workshop is to bring together experts in probability and representation theory to permit a fruitful exchange of ideas in this fertile interdisciplinary area.
Anupam Singh
Gaussian elimination in split classical groups
The classical Gaussian elimination writes an invertible matrix as a product of elementary matrices and a diagonal matrix. We present a similar algorithm for Orthogonal and Symplectic groups. We begin with describing analogue of elementary matrices and row-column operations and describe the algorithm. This is joint work with Ayan Mahalanobis.
Sushil Bhunia
Computing spinor norm in the split orthogonal group
Using Wall's theory we describe how to compute spinor norm using Gaussian elimination algorithm for the split orthogonal group in odd characteristic.
Prahlad Shinde
MOR cryptosystem and Chevalley groups
Study of MOR cryptosystem using finite Chevalley groups in odd characteristics
Amritanshu Prasad
Kronecker Coefficients, Integer Arrays and the RSK Correspondence
A Kronecker coefficient counts the number of times a representation of a symmetric group occurs in the tensor product of two others. Finding a fast algorithm to determine when a Kronecker coefficient is positive is an open problem. There has been an increased interest in this problem over the last few years as it comes up in the geometric approach to the complexity conjecture P = NP due to Mulmuley and Sohoni. I will explain how a higher dimensional analogue of the Robinson-Schensted-Knuth correspondence relates Kronecker coefficients to the problem of counting the number of integer arrays with specified slice sums.
Pooja Singla
Generating a random permutation with random transpositions
Consider the problem of n cards labelled 1 through n. Suppose two integers i and j are chosen independently and uniformly between 1 and n. The cards labelled i and j are switched. If many such transpositions are made the row of cards will tend to appear in random arrangement. Then question is how many steps are required until the deck is well mixed up (i.e. the permutation is close to random)? Diaconis and Shahshahani used tools of representation theory of symmetric groups to prove that at least 1/2(n log n) steps are required before the deck will be well mixed up for large n. I will explain ideas of their proof and few related problems.
Steven Spallone
The Circular Unitary Ensemble
Diaconis and Shahshahani used the representation theory of the unitary group to understand the distribution of the trace of a large random unitary matrix. We explore these ideas in the lecture.
K N Raghavan
Littelmann's LS paths and the Pitman transform
The talk will be a report on (the more representation theoretic parts of) the paper of Biane, Bougeral, and O'Connell titled "Littelmann Paths and Brownian Paths" (arXiv 0403171). We will define the notion of a (generalized) Pitman transform and show how this relates to Littelmann's root operators, whose relevance to the representation theory of reductive algebraic groups will be recalled. The speaker hopes that the audience will help him understand the more probabilistic parts of the paper.
U. K. Anandavardhanan
Symplectic local root numbers for GL(2)
We will start with an observation due to Diaconis and Shahshahani relating the transition matrix associated to a probability measure on a finite group to its Fourier transforms at irreducible representations, and then indicate a potential application of this observation to certain questions about local root numbers associated to infinite dimensional representations of p-adic GL(2). This latter work is part of a joint project with Dipendra Prasad.
Uday Bhaskar Sharma
Asymptotic for Counting Representations of Polynomial Algebras over Finite Fields
Let c(n,k,q) be the number of isomorphism classes of n-dimensional representations of the polynomial algebra F_q[x_1,...,x_k] (which is the same as similarity classes of commuting k-tuples of n x n matrices over the finite field F_q). We will show that c(n,k,q) is asymptotically q^(([n^2/4]+1)k) as a function of k.
Amit Kulshrestha
Quadratic maps and special 2-groups
Special 2-groups are the 2-groups for which the commutator subgroup, Frattini subgroup and the centre coincide to an elementary abelian group. These groups can be described in terms of quadratic maps between vector spaces over fields of characteristic 2. In this talk we describe an algorithm that takes a quadratic map as input and outputs the complex character table and conjugacy classes of a real special 2-group. This is a joint work with Dilpreet Kaur. An attempt will be made to explore if quadratic maps can be used to study random walks on special 2-groups.
Arvind Ayyer
Random-to-random shuffling for partially ordered sets
The random-to-random shuffle involves picking a card from a (uniformly) random position i and inserting it into a (uniformly) random position j. The cards in positions between i and j are thus shifted by 1. We are interested in understanding how long it takes for the deck to be well-mixed. One measure of this mixing is the second-largest eigenvalue of the transition matrix. An explicit formula for this was conjectured circa 1995 and a proof has been announced recently. We generalise this conjecture to linear extension of partially ordered sets. This is joint work with Anne Schilling and Nicolas Thiery.
Manjunath Krishnapur
An exposition on the split-merge chain on random partitions and the
Poisson-Dirichlet distribution
The cycle sizes of a uniform random permutation give rise naturally to a probability distribution on partitions of the unit interval denoted PD_n. Consequently, the random transposition Markov chain on permutations gives rise to a 'split-merge' markov chain on the space of random partitions of the interval and PD_n is the unique stationary distribution of this Markov chain.
As n-->\infty, there is a limiting Markov chain on all partitions of the unit interval and PD_n converges to a probability distribution PD on this space. Vershik conjectured and Diaconis, Mayer-Wolf, Zeitouni and Zerner proved that PD is the unique stationary distribution for this Markov chain.
We give an exposition of this work and try to emphasize the part that uses representation theory of the symmetric group.