Talks(Abstracts)

Speaker: P. Amrutha

Title:   Cyclic characters of alternating groups

Abstract:    

The cyclic characters of a group G are the characters induced from the cyclic subgroups of G. In the case of classical Coxeter groups, Kraskiewicz and Weyman worked out the decomposition into irreducible characters of characters induced from the cyclic subgroup generated by a Coxeter element. Jöllenbeck and Schocker gave a general approach for the case of symmetric group S_n by considering the cyclic group generated by any element of S_n. The cyclic characters of S_n are described in terms of a statistic on the Young tableaux called the multi-major index. In this talk, we will see a description of the cyclic characters of alternating groups. This is joint work with Amritanshu Prasad and Velmurugan S.

Speaker: Prashant Arote

Title:   Prasad’s Conjecture about dualizing involution

Abstract:    

Let G be a connected reductive group defined over a finite field 𝔽q with corresponding Frobenius F. Let 𝑙G(subscript) denote the duality involution defined by D. Prasad under the hypothesis 2H¹(F, Z(G)) = 0, where Z(G) denotes the center of G. In this talk, we will show that, for each irreducible character ρ of G^F, the involution 𝑙G(subscript) takes ρ to its dual ρ^∨ if and only if for a suitable Jordan decomposition of characters, an associated unipotent character uρ(subscript) has Frobenius eigenvalues ± 1. As a corollary, we get that if G has no exceptional factors and satisfies 2H¹(F, Z(G)) = 0, then the duality involution 𝑙G(subscript) takes ρ to its dual ρ^∨ for each irreducible character ρ of G^F. Our results resolve a finite group counterpart of a conjecture of D. Prasad. This is a  Joint work with Manish Mishra.

Speaker: Arvind Ayyer

Title:   The multispecies totally asymmetric long-range exclusion process and Macdonald polynomials

Abstract:    

The multispecies totally asymmetric long-range exclusion process (mTALREP) is an interacting particle system with multiple species of particles on a finite ring where the hopping rates are site-dependent. (The homogeneous variant on Z is also known as the Hammersley–Aldous–Diaconis process.) In its simplest variant with a single species, a particle at a given site will hop to the first available site clockwise. We show that the partition function of this process is intimately related to the classical Macdonald polynomial and to the multispecies totally asymmetric simple exclusion process (TASEP). We also show that well-known families of symmetric polynomials appear as expectations in the stationary distribution of important observables. This is joint work with James Martin and Omer Angel.

Speaker: Dibyendu Biswas

Title:   Branching Law for Classical Groups

Abstract:    

In this talk, we will talk about some of the branching problems: Given a pair G ⊃ H of reductive groups and an irreducible representation π  of G, find the decomposition π |H(subscript) into irreducible representations. We will discuss the known result for the case Sp(2n) ⊃ Sp(2n-2). Then analogously, we will talk about the case for SO(n) ⊃ SO(n-2). 

Speaker: Pralay Chatterjee

Title:   Lower dimensional Betti numbers of homogeneous spaces of Lie groups

Abstract:    

We will talk about certain explicit descriptions of lower dimensional Betti numbers of homogeneous spaces of compact Lie groups. We will begin by recalling some of the earlier works on this subject and sketch our motivation. We will then describe our results and some of the applications in special cases. If time permits we will also sketch some of the proofs. This is joint work with Indranil Biswas and Chandan Maity.

Speaker: Anuradha Garge

Title:   Gow-Tamburini type generation of the special linear group for some special rings

Abstract:    

A result of Hurwitz says that the special linear group of size at least three over the ring of integers of an algebraic number field is finitely generated. Since the special linear group of size at least three over the ring of integers is not a finite simple group, we expect that it has more than two generators. Motivated by the work of Gow and Tamburini for the ring of integers, we provide a set of three generators (called the Gow Tamburini generators), depending upon the conditions on the ring, for the generation of the special linear group of size atleast three. 

Speaker: Krishnendu Gongopadhyay

Title:   Reciprocity in Hecke Groups

Abstract:    

An element g in a group G is called reciprocal if there exists h ∈ G such that g^−1 = hgh^−1. The reciprocal elements are also known as ‘real elements’ or ‘reversible elements’ in the literature. We classify the reciprocal elements and parametrize the reciprocal classes in the Hecke groups.  This is joint work with Debattam Das. 

Speaker: Sagar B. Kalane

Title:   Free and discrete groups generated by two parabolic maps

Abstract:    

In this talk, we consider a group generated by two unipotent parabolic elements of SU(2, 1) with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite R-circle in the Heisenberg group. This is joint work with Prof. John Parker. 

Speaker: Chayan Karmakar

Title:   Character of G_2 at the unique conjugacy class of order 2

Abstract:    

The simple algebraic group G_2(C) has a unique conjugacy class C_2 of order 2 whose centralizer is isomorphic to  [SL_2(C) x SL_2(C)]/ ± 1. We calculate the character of any irreducible finite-dimensional representation Π {k,l} of G_2 of highest weight kΩ_1 + l Ω_2 (where Ω_1 and Ω_2 are the fundamental representations of G_2) at C_2 and interpret the result in terms of the dimension of a suitably defined irreducible representation of [SL_2(C) x SL_2(C)]/± 1 associated to the representation Π{k,l} of G_2. The irreducible representation of [SL_2(C) x SL_2(C)]/± 1 depends on the congruences of k,l mod 2. In particular, the character of the representation Π{k,l}; at C_2, hence the irreducible representation of   [SL_2(C) x SL_2(C)]/ ± 1, is zero if and only if both k,l are odd.

Speaker: Himanshi Khurana

Title:   Twisted Jacquet Module

Abstract:    

In the first part of the talk, we will briefly discuss the work of Prasad on the twisted Jacquet module to motivate our problem. In the second part of the talk, we will describe the structure of the twisted Jacquet module π{N, Ψ} of an irreducible cuspidal representation π of GL(2n, 𝔽q). 

Speaker: Didier Lesesvre

Title:   Uniform Counting of automorphic representations

Abstract:    

Automorphic representations are fascinating objects, covering as various settings as elliptic curves, modular forms, eigenwaves of PDEs or Galois representations. The Langlands program seeks to study them, but conjectures remain widely open. In spite of understanding a single automorphic form, a powerful approach is to consider them in family, and this is the spirit of trace formulas : a way of doing harmonic analysis on noncommutative groups. Trace formulas are bridges between spectrum of an operator, geometry of a variety and arithmetic of a group. I will try to introduce the basic spirit of the trace formulas, focus on some simple cases in particular in the group-theoretic setting, show how they sit in the main challenges on automorphic forms, and how they open unsolved questions on group theory and representations of p-adic groups.

Speaker: Deepkumar Makadiya

Title:   Automorphisms of Chevalley Groups

Abstract:    

In this talk, we discuss the automorphisms of the Chevalley groups over a commutative ring with unity. We present the work of Steinberg, Humphreys, and Bunina, who respectively classified the automorphisms in the finite (over fields), infinite (over fields), and general (over rings) cases. We explain the result that any automorphism of the adjoint Chevalley group can be expressed as a composition of these four types: inner, central, graph, and ring automorphisms. This leads to a comprehensive classification of all automorphisms within the adjoint Chevalley group. If time permits, we will discuss the joint work with Shripad Garge on the automorphisms of twisted Chevalley groups.

Speaker: Neha Malik

Title:   The Subalgebra of Stiefel-Whitney Classes for Finite Symplectic Groups

Abstract:    

Orthogonal representations π of a finite group G have a sequence of cohomological invariants wi(π) living in the mod 2 group cohomology, called Stiefel-Whitney Classes (SWCs). We have computed these classes for several finite groups of Lie type. Such calculations can answer interesting questions like, what is the subalgebra of H* (G, Z/2Z) generated by SWCs of all orthogonal π? Is it the whole group cohomology ring? Based on recent work with Prof. Steven Spallone, this talk will give an overview of some of our results for finite symplectic groups.

Speaker: Sridhar Narayanan

Title:   Restriction Coefficients - Survey and recent results

Abstract:    

The restriction problem is an open problem in algebraic combinatorics. In this talk we define the problem, survey its history, and present some edge-case results obtained by us (Shraddha S., Amritanshu P.,  Digjoy P.).

Speaker: Ashish Mishra

Title:   Branching rule for complex reflection groups

Abstract:    

The branching rule for the family of symmetric groups is well known and plays an important role in the study of representation theory of partition algebras. The family of symmetric groups, Sn, forms a subfamily of the family of complex reflection groups G(r, p, n), where r, p and n are natural numbers such that p is a divisor r. While studying the representation theory of partition algebras for complex reflection groups, we need to consider the “Sn−1-analog” in           G(r, p, n), i.e., a subgroup of G(r, p, n) which plays a role analogous to the role played by Sn−1 ⊂ Sn. In this talk, using a confluence of Okounkov–Vershik approach, Clifford theory and higher Specht polynomials, we study the branch-ing rule from G(r, p, n) to the Sn−1-analog in G(r, p, n). The results presented in this talk form parts of a joint work with Shraddha Srivastava.

Speaker: Shiv Prakash Patel

Title:   Euler-Poincare characteristic and covering groups

Abstract:    

Let F be a p-adic field. Let GL_{n}(F) be the subgroup of GL_{n+1}(F) as embedded in the first n x n block.  We will discuss the Euler-Poincare characteristic, formulated by Dipendra Prasad, for the restriction of representations of GL_{n+1}(F) to the subgroup GL_{n}(F). Let G be the n-fold Kazhdan-Patterson cover of GL_{n+1}(F) and H be the inverse image of GL_{n+1}(F) in G. We make some analogous remarks for restrictions of representations of G to H.

Speaker: Basudev Pattanayak

Title:   Hecke algebras of GL(2) over P-adic division algebra

Abstract:    

The main question will be addressed in this talk: To what extent does the representation theory of GL(2, D), or the Hecke algebra H[GL(2, D)] depend on the division algebra D over a local field? Let D and D’ be two division algebras defined over non-archimedean local fields of characteristic zero. Then, using Bushnell-Kutzko type theory and Bernstein decomposition of Hecke algebra, we will show that the Hecke algebra H[GL(2, D)] is Morita equivalence to the algebra H[GL(2, D’)]. As a consequence, we show that the category of smooth representations of GL(2, D) does not depend on D. This is joint work with Amiya Mondal.

Speaker: Dipendra Prasad

Title:   Degenerate Whittaker models

Abstract:    

Whittaker models have played a large role in the representation theory of reductive groups over finite, local and global fields. They are not available for all representations. There are the degenerate Whittaker models which are available more generally, usually studied when there are no Whittaker models, however, they have interest even when there is a Whittaker model. The lecture will be an exposition of some results, some due to me and others due to others.

Speaker: Radhika M. M.

Title:   Lattices in ℝ^n ⋊ SL_2(ℝ)

Abstract:    

We will discuss the existence of cocompact lattices in V ⋊ SL_2(ℝ) for any representation V of SL_2(ℝ). The talk is based on joint work with Sandip Singh.

Speaker: K.N. Raghavan

Title:   Simple Procedures for Keys

Abstract:    

This talk is based on the following preprint (written jointly with Mrigendra Singh Kushwaha and Sankaran Viswanath):  https://arxiv.org/abs/2302.08279

Demazure submodules of an irreducible polynomial representation of the general linear group GL(n) are indexed by permutations of n letters. Fix a partition p with at most n parts (these parametrize irreducible polynomial representations of GL(n)) and a permutation w. Let D(p,w) denote the corresponding Demazure submodule of the irreducible polynomial representation V(p). Semi-standard tableaux of shape p in which every entry is at most n form a combinatorial model for V(p).  In particular, the character of V(p) is encoded by such tableaux. To each such tableaux Lascoux and Schuetzenberger associated two permutations called  "right key" and  "left key". Keys derive their interest from the following fact (and its "left" or "opposite" analogue):   the subset of such tableaux whose right key is <=w forms a combinatorial model for the Demazure submodule D(p,w). Given the importance of keys,  there are many procedures available for computing them. In the preprint mentioned above, we give new, simple procedures. (In "standard monomial theory" developed by Lakshmibai, Musili, and Seshadri,  keys are obtained by performing optimal "Deodhar lifts". In Littlemann's path model theory, keys are the initial and final directions of Lakshmibai-Seshadri paths. Our procedures for keys are based in turn upon procedures for performing optimal Deodhar lifts.)

Speaker: M. S. Raghunathan

Title:   The Chevalley Basis of a Semi-simple Lie Algebra

Abstract:    

I will sketch proof of the existence of a Chevalley basis for a semi-simple Lie Algebra. The proof is somewhat different from that of Chevalley. The Weyl group plays an important role in the proof.

Speaker: Uday Bhaskar Sharma

Title:   A Frobenius-Type Formula for Compact Lie Groups

Abstract:    

Let G be a finite group, and α: G x G → G denote the commutator map. The cardinalities of the fiber α^{-1}(g) can be computed by a formula due to Frobenius, using the character values χ(g) of irreducible characters χ of G. We derive the generalized formula for connected compact lie groups, orthogonal groups, as well as FC groups. This is a joint work with Shripad Garge.

Speaker: Anupam Kumar Singh

Title:   Surjectivity of word maps on matrix algebras

Abstract:    

Given a polynomial in d non-commuting variables, one defines a map on a central simple algebra A and wants to understand the nature of the image, like Kaplansky-L'ovo conjecture. There has been tremendous progress for analogous questions for a word w in d variables which gives rise to a map on a group G defined by evaluations, called word maps. In this talk, we will see some of the results proved in this direction and the kind of questions being studied. Such questions are motivated by the Waring problem in number theory and Ore's conjecture in group theory. In a recent work jointly with Prachi Saini and Saikat Panja, we studied the surjectivity of the diagonal word and a multilinear map given by the product of commutators. 

Speaker: Shashank Vikram Singh

Title:   Arithmeticity of Some Symplectic Hypergeometric Groups

Abstract:    

In this talk, we will talk about the arithmeticity of some symplectic hypergeometric groups (SHGGs). We will discuss a criterion that helps in determining the arithmeticity of some of the SHGGs. To verify the criterion,  a computer-based program is developed. We will see the working of the program. Using the criterion and the program, we have determined the arithmeticity of some of the SHGGs of degree six. It is joint work with Bajpai, Dona, and Singh.

Speaker: Steven Spallone

Title:   Central Extensions and Cohomology

Abstract:    

Let A be a discrete abelian group. When G is a finite group, there is a well-known natural correspondence between central extensions of G by A and the quadratic group cohomology H^2(G, A). From this, one can systematically find cohomological criterion for lifting problems. In this talk, we discuss an analogue of this correspondence when G is a Lie group. This talk represents joint work with Rohit Joshi. 

Speaker: Sandeep Varma

Title:   Transferring some character values across close local fields

Abstract:    

If G is a split reductive group over Z, and F and F' are non-archimedean local fields that are close in a suitable sense, then there is a bijection, defined by Kazhdan, between irreducible smooth representations of G(F) and G(F') that have nonzero vectors fixed under appropriate compact open subgroups. Radhika Ganapathy proved that this transfer of representations preserves several important properties of arithmetic interest. We will consider split classical groups and study the behavior of certain character values with respect to this transfer. This is joint work with A. M. Aubert. Please note that the results are tentative since this is ongoing work.

Speaker: Venketasubramanian, C. G.

Title:   Structure of Twisted Jacquet modules for principal series representations

Abstract:    

Let F be a non-archimedean local field. Let G denote the symplectic group Sp_4(F) and let π be a principal series representation of G parabolically induced either from the Siegel parabolic subgroup P or its Klingen parabolic subgroup Q. In this talk, we shall discuss about ongoing work (joint with Sanjeev Kumar Pandey) on the structure of the twisted Jacquet module of π taken with respect to a character of the unipotent radical N of P. 

Speaker: Mahendra Verma

Title:   Representations of GL(n, D) with Symplectic period

Abstract:    

In this talk, we consider the question of determining when an irreducible representation of GL(n, D) where n = 3, 4, admits a symplectic period, i.e., when such a representation has a linear functional invariant under Sp(n, D), where D is a quaternion division algebra over a non-archimedean local field k and Sp(n, D) is the unique non-split inner form of the symplectic group Sp(2n, k). This is joint work with  Hariom Sharma.