Algebra Seminar
Speaker: Prof. Sankaran Viswanath (IMSc, Chennai)
Host: Krishnan Sivasubramanian
Title: A tale of two representations
Time, day and date: 11:00:00 AM, Monday, May 26
Venue: Ramanujan Hall
Abstract: We describe two well-known families of representations of the Lie algebra sl_n of traceless square matrices of size 'n' - finite-dimensional irreducible representations 'V' and tensor products of fundamental representations 'E'. Both are indexed by partitions and admit combinatorial models in terms of variants of Young tableaux. The latter module also admits an action of the infinite dimensional Lie algebra of traceless matrices whose entries are polynomials in a single variable. This ("affine") point of view gives yet another combinatorial model for 'E', in terms of Gelfand-Tsetlin patterns with a partition overlay. In this talk, we show how the finite and affine combinatorial models of 'E' are in fact related by "functorial" bijections, which respect natural projection and branching maps. We will also visualise this bijection in terms of ensembles of coloured lattice paths on a square lattice, and provide a "Proof without words" of sorts. This is based on joint work with Aritra Bhattacharya and TV Ratheesh (arXiv:2412.00116).

Combinatorics seminar
Speaker: Dr. Prasant Singh (IIT Jammu)
Host: Sudhir R. Ghorpade
Title: Hyperplane Sections of Determinantal Varieties of Symmetric Matrices over Finite Fields
Time, day and date: 4:00:00 PM, Tuesday, May 27
Venue: Ramanujan Hall
Abstract: Let X = (Xij ) be a m × m generic symmetric matrix whose entries are independent indeterminates over a field F. The symmetric determinantal variety St = St(m) is given by vanishing of (t + 1) × (t + 1) minors of X. This variety is defined over any finite field Fq and has many Fq-rational points. This makes it a useful object from the point of view of applications to coding theory. An explicit formula for the number of Fq-rational points St was determined by Carlitz (1954) in a special case and by MacWilliams (1969) in the general case. Partly from the viewpoint of applications, one is also interested in the following questions:

(i) What are the possible values of |St ∩ H(Fq)|, where H is a Fq-rational hyperplane in the projective space P ( m+1/2 )−1?

(ii) What is the maximum possible value of |St ∩ H(Fq)|, where H varies over the hyperplanes as in (i) above?

In this talk, we will address both the problems above and answer them in the cases when the Fq is of odd characteristic case. This is a joint work with Peter Beelen and Trygve Johnsen.

Link to the abstract: https://drive.google.com/open?id=11m-HFafFkmCOTv6oyozxsL2em81TKJYn

Speaker: Deep Makadiya (IIT Bombay, Mumbai)
Host: Shripad Garge
Title: Some properties of twisted Chevalley groups
Time, day and date: 11:30:00 AM, Wednesday, May 28
Venue: Ramanujan Hall
Abstract: Let $R$ be a commutative ring satisfying mild conditions. Let $G_{\pi,\sigma} (\Phi, R)$ denote a twisted Chevalley group over $R$, and let $E'_{\pi, \sigma} (\Phi, R)$ denote its elementary subgroup. This thesis discussed some problems in the theory of twisted Chevalley groups.

The first problem concerns the normality of $E'_{\pi, \sigma} (\Phi, R, J)$, the relative elementary subgroups at level $J$, in the group $G_{\pi, \sigma} (\Phi, R)$ and the second problem addresses the classification of the subgroups of $G_{\pi, \sigma}(\Phi, R)$ that are normalized by $E'_{\pi, \sigma}(\Phi, R)$. This classification provides a comprehensive characterization of the normal subgroups of $E'_{\pi, \sigma}(\Phi, R)$. Lastly, the third problem investigates the normalizers of $E'_{\pi, \sigma}(\Phi, R)$ and $G_{\pi, \sigma}(\Phi, R)$ in the bigger group $G_{\pi, \sigma}(\Phi, S)$, where $S$ is a ring extension of $R$. We prove that these normalizers coincide. Moreover, for groups of adjoint type, we show that they are precisely equal to $G_{\pi, \sigma}(\Phi, R)$.

Statistics and Probability seminar
Speaker: Arindam Chatterjee (ISI Delhi)
Host: Debraj Das
Title: Statistical inference using network sampling in a sparse Stochastic Block Model (SBM) setup
Time, day and date: 4:00:00 PM, Thursday, May 29
Venue: Ramanujan Hall
Abstract: We consider the problem of predicting subgraph counts and the clustering coefficient of a large population network on $N$ nodes using a network sampling scheme. The population network is assumed to be generated from a SBM with edge probabilities decaying to zero at the rate $N^{-\beta}$, for some $\beta\in [0,2]$. We study Bernoulli node sampling (with a fixed node selection probability $p\in (0,1)$), followed by either induced or ego-centric subgraph formation. Given a fixed target subgraph $H$ with $R$ nodes and $T$ edges, we show that the limiting distribution of the scaled and centered sample based subgraph count is asymptotically normal, if $\beta\in [0, R/T)$, and the limit law is Poisson, if $\beta = R/T$. Using a multivariate version of this result we obtain limit laws for the sample based clustering coefficient. As a follow up, for specific choices of subgraphs, we also investigate the case where $p = p_N$ is allowed to decay to zero at a certain rate. We find surprising differences between the effects of induced and ego-centric sampling in this setting.

(This is an ongoing work with my PhD student, Anirban Mandal)