Representations of Groups and Algebras

Anupam Singh

IISER Pune

Title: Word maps and chirality on groups

Abstract: Given an element of the free group on d generators, called a word (say w), and a group G, we can define a map by evaluation. Such maps are called word maps. In the last 3 decades, several great results, called Waring-like results, for finite simple groups have been obtained. In general, the main question in the subject has been to understand the image w(G). In this talk, we try to understand when the image w(G) is symmetric under inversion for finite groups of Lie type.

Apoorva Khare

IISc Bangalore

Title: Log-concavity in algebra and combinatorics, and for weight multiplicities of parabolic Vermas

Abstract: We will begin by discussing various instances of log-concavity in algebra and combinatorics, including involving symmetric functions. This includes several works of June Huh with coauthors; in particular, we will discuss his result with Matherne-Meszaros-St.Dizier (2022), that the weight-space multiplicities of any finite-dimensional simple sl(n+1)-module V, form a log-concave sequence along every root direction. In forthcoming work with Matherne and St.Dizier, we extend this to all parabolic (or first-order) Verma modules V, and then explore the situation for higher-order Verma modules V.

Arvind Ayyer

IISc Bangalore

Title: How large is the character degree sum compared to the character table sum for a finite group?

Abstract: In 1961, Solomon proved that the sum of all the entries in the character table of a finite group does not exceed the cardinality of the group. We state a different and incomparable property here -- this sum is at most twice the sum of degrees of the irreducible characters. Although this is not true in general, it seems to hold for "most" groups. We establish the validity of this property for symmetric, hyperoctahedral and demihyperoctahedral groups. Using these techniques, we are able to show that the asymptotics of the character table sums is the same as the number of involutions in these groups. We will also derive generating functions for the character tables sum for these groups as infinite products of continued fractions. This is joint work with D. Paul and H. K. Dey (arXiv:2406.06036).

B. Ravinder

IIT Tirupati

Title: Bases for generalized Weyl modules in type $A$

Abstract: Classical local Weyl modules for a current algebra are $\mathbb{Z}$-graded modules where the zeroth-graded pieces are the irreducible representations of the underlying simple Lie algebra. Generalized Weyl modules are also $\mathbb{Z}$-graded, but their zeroth-graded pieces are the Demazure modules. Chari and Loktev provided a basis for the local Weyl modules in type $A$. In this talk, we will present a basis for certain generalized Weyl modules, based on ongoing work with Samiran Nayek.

Chudamani Pranesachar Anil Kumar

Krea University

Title: On the vanishing criterion for the mod p cohomology groups of the automorphism group of a finite abelian p-group

Abstract: Fields medalist D. Quillen has given a vanishing range for the mod $p$ cohomology ($p$ a prime) of the finite general linear group in one of his annals papers ($MR0315016$: \url{https://www.jstor.org/stable/1970825}) which was further improved by E.M. Friedlander and B.J. Parshall ($MR0722727$: \url{http://eudml.org/doc/143063}). The automorphism group of a finite abelian $p$-group associated to a partition can be thought of as a generalization of the finite general linear group which is the automorphism group of the finite elementary abelian group. However, nothing much is known about the mod $p$ cohomology of the automorphism group of a finite abelian $p$-group except when the abelian $p$-group is elementary. Even in the elementary abelian case only a vanishing range for the cohomologies is known. In 2018, S.~Galatius, A.~Kupers and O.~R.~Williams have extended the vanishing range for the mod $p$ cohomology of $GL_n(\mbb{F}_{q})$ when $q=p^r>4$\ (arXiv:1810.11931v1). So obtaining a vanishing criterion for the first and second cohomology groups in the general case itself is a new result in this direction. In this talk, we mention without proof that for an odd prime $p$ the first cohomology group vanishes if and only if the difference between the successive parts of the partition is at most $1$ with the last part being equal to $1$. We also prove that for an odd prime $p\neq 3$ the second cohomology group vanishes if and only if the difference between the successive parts of the partition is at most $1$ with the last part being equal to $1$. This is based on joint work (arXiv: 2112.10610) with Dr. Soham Swadhin Pradhan, University of Haifa, Israel. Finally we mention the vanishing range conjecture for the mod $p$ cohomology of the automorphism group of an abelian $p$-group. The talk will be accessible to motivated, advanced undergraduate students and doctoral students (especially to those whose areas of interest include cohomology of finite groups). The topic may be of interest to others as well because the cohomology of finite groups is a very vast subject which has its interactions with other areas such as group theory, representation theory, homological algebra, number theory, K-theory, classifying spaces, group actions, characteristic classes, homotopy theory.

Digjoy Paul

IISc Bangalore

Title: The Immersion Poset on Partitions

Abstract: We introduce the immersion poset on partitions, defined by the monomial-positivity of Schur polynomial differences. Relations in this poset characterize when irreducible polynomial representations of general linear groups form an immersion pair, as defined by Prasad and Raghunathan. In joint work with Anne Schilling and her graduate students, we investigate covering relations and maximal elements of the immersion poset. By proving the Schur-positivity of certain power sum symmetric polynomials associated with lower intervals in the poset, we address a question posed by Sheila Sundaram: "Which subsets of columns in the character table of the symmetric group yield non-negative row sums?"

Geetha Thangavelu

IISER Thiruvananthapuram

Title: Permutation modules of the walled Brauer algebras

Abstract: The walled Brauer algebras arises in the context of Schur-Weyl duality where the general linear group Gl_n acts on the mixed tensor space of its natural representation V and its dual. In this talk, we will see a construction of permutation modules of direct product of symmetric groups and the walled Brauer algebras which is similar to the construction of permutation modules by Hartmann, Henke, Koenig and Paget for cellularly stratified algebras as well the construction from Inga Paul’s paper. We prove that if the characteristic of the field is other than 2 and 3, the permutation modules of walled Brauer algebras can be written as a direct sum of indecomposable Young modules. This result is also extended to other diagram algebras.This is a joint work with Sulakhana Chowdhury.

Jyotirmoy Ganguly

GITAM University Bangalore

Title: Multiplicity-Free Tensor Products

Abstract: A finite group G is said to have multiplicity-free tensor products if, for all irreducible representations U, V of G, the internal tensor product of U and V has a multiplicity free decomposition into irreducible representations of G. We investigate which finite groups have multiplicity-free tensor products.

Krishanu Roy

SRM University-AP

Title: Partial order of dominant weights of affine Kac–Moody algebras

Abstract: The dual space of the Cartan subalgebra in a Kac–Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple roots. In this talk, we will look at the subposet formed by dominant weights in affine Kac–Moody algebras. We will look at an explicit description of the covering relations in this poset.

KV Subrahmanyam

CMI

Title: TBA

Abstract:TBA

Mrigendra Singh Kushwaha

Delhi University

Title: Gelfand-Tsetlin Crystals of Kostant-Kumar modules

Abstract: We give Gelfand-Tsetlin crystals for the Kostant-Kumar modules for the finite simple Lie algebra of type A. Kostant-Kumar modules are cyclic submodules of the tensor product of two irreducible highest weight modules of a symmetrizable Kac-Moody Lie algebras. In this case (type A), we also provide a polytopal model for Kostant-Kumar modules in terms of BiKogan faces.

Pampa Paul

Presidency University

Title: Borel-de Siebenthal positive root systems

Abstract: Let $G$ be a simple equi-rank Lie group with Lie algebra $\frak{g}_0$ and $\frak{g} = \frak{g}_0^\mathbb{C}.$ Borel and de Siebenthal have proved the existence of a positive root system of $\frak{g}$, known as a Borel-de Siebenthal positive root system, in which there is exactly one non-compact simple root and the coefficient of that non-compact simple root in the highest root is exactly one or two. In this talk, we will describe all Borel-de Siebenthal positive root systems of $\frak{g}$ and the number of equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ with a fixed infinitesimal character.

Parameswaran Sankaran

CMI

Title: Picard groups of certain compact complex manifolds

Abstract: The set of all isomorphism classes of holomorphic line bundles over a complex manifold forms a group under tensor product. We shall discuss some examples and state some recent results for a class of complex manifolds which are either holomorphically parallelizable or are compact locally Hermitian symmetric spaces. The talk is based on joint work with Pritthijit Biswas.

Pooja Singla

IIT Kanpur

Title: TBA

Abstract:TBA

Preena Samuel

IIT Kanpur

Title: Invariant theory of superspaces

Abstract: In this talk we revisit an old problem in invariant theory of finding a generating set for the ring of invariants of n × n-matrices under the action of the general linear group and other classical groups. We recast it in the setting of superspaces and give an answer to this question and a related question of invariants of mixed tensor spaces of a finite dimensional vector superspace V over C.

Senthamarai Kannan

CMI

Title: Cohomology of line bundles on horospherical Schubert varieties

Abstract: Let $G$ be a semisimple simply connected algebraic group over $\mathbb{C}$. $T$ a maximal torus of $G$, $B$ a Borel subgroup of $G$ containing. $R$ denotes root system of $G$ relative to $T$ and $R^{+} be the positive roots of $G$ relative to ($B$ , $T$). Let $W=N_G{T)$ be the Weyl group of $G$ relative to $T$. For a character $\lambda$ of $T$ and an element $w\in W$, the dot action is defined by $w.\lambda=w(\lambda+rho)-\rho$, Where $\rho$ denotes the half sum of all positive roots. Let $G/B$ be the full flag variety of all Borel subgroups of $G$. We have the celebrated Borel-Weil-Bott’sTheorem. Let $\lambda$ be. character of $T$. Then we have 1. If $\lambda+\rho$ is singular, then $H^{i}(G/B, L_{\lambda})=0$ for all $i\in \mathbb{Z}_{\geq 0}. 2, If $\lambda + \rho$ is non singular, then there is a unique $\phi \in W$ such that $\phi.lambda$ is dominant. Further, $H^{l(\phi)}(G/B, L_{\lambda})=H^{0}(G/B, L_{\phi.\lambda})$. So, it is a natural to ask whether given a Schubert variety $X(w):=\overline {BwB/B}$, what are all $\phi$ such that for generic weights $\lambda$ in the $\phi$ chamber only cohomology is non zero. In this direction, there are many results obtained by various authors . But the uniqueness question is not yet answered in complete detail. I will explain these results briefly if time permits. Now, let $X(w)$ be a Schubert variety. Let $P$ be the stabiliser of $X(w)$ in $G$. Let $Q\subset P$ be a parabolic subgroup of $G$ containing $B$, and let $L$ be the Levi subgroup of $Q$ containing $T$. Then we give a criteria for when $X(w)$ is a $L$- horospherical Schubert variety. Further along the direction of uniqueness of non vanishing of cohomologies, we give a criterion for $L$- horospherical Schubert varieties in terms of the combinatorics between $w$ and $\phi$. This is based on an on going joint work with Mahir Bilen Can and Pinakinath Saha.

Sachin Subhash Sharma

IIT Kanpur

Title: Weyl modules for twisted toroidal Lie algebras

Abstract: In this talk, we extend the notion of Weyl modules for twisted toroidal Lie algebra T (µ). We prove that the level one global Weyl modules of T (µ) are isomorphic to the tensor product of the level one representation of twisted affine Lie algebras and certain lattice vertex algebras. As a byproduct, we calculate the graded character of the level one local Weyl modules of T (µ).

Santosha Kumar Pattanayak

IIT Kanpur

Title: Variety of commuting matrices and a higher dimensional Chevalley restriction theorem.

Abstract: The variety of d-tuples of commuting n x n matrices is an object of great interest in Mathematics. It has applications in algebraic geometry, representation theory, symplectic geometry and complexity theory. However, its geometry is complicated and surprisingly many simple looking conjectures are still open. We will discuss the geometry of the commuting variety via a higher dimensional Chevalley restriction theorem.

Shripad M. Garge

IIT Bombay and HRI

Title: Around McKay conjecture

Abstract: It is a conjecture of John McKay that p' degree irreducible characters of a finite group are the same in number as that of the normlizer of a Sylow p-subgroup of the group. This conjecture is proved for p = 2 by Malle and others. We investigate if there is a nice bijection between the two sets of characters for finite Coxeter groups. This is a joint work with P. Amrutha.

Shyamashree Upadhyay

IIT Guwahati

Title: Schubert varieties in the Grassmannian and the bounded RSK correspondence

Abstract: In a paper by Kodiyalam and Raghavan, they provide an explicit combinatorial description of the Hilbert function of the tangent cone at any point on a Schubert variety in the Grassmannian. The main idea of their work is a bijection between two combinatorially defined sets. In this talk, we discuss the fact that this bijection is in fact a bounded RSK correspondence.

Sridhar Poojyam Narayanan

TIFR

Title: Trees, parking functions and ideals

Abstract: This is a survey of some combinatorial and algebraic objects defined on graphs, highlighting the connections between them and outlining some interesting areas for exploration.

Tanusree Khandai

IISER Mohali

Title: Fusion Product Modules for Current Algebras of type $A_2$

Abstract: Fusion product modules for current Lie algebras were introduced by Feigin and Loktev in 1999. To better understand the structure of these modules, Chari and Venkatesh introduced a class of modules in 2015, now referred to as CV modules. In this talk, I will present results from a joint work with Shushma Rani, where we use CV modules to obtain a graded character of the fusion product of two irreducible $\mathfrak{sl}_3​[t]$-modules.

Uday Bhaskar Sharma

UPES, Dehradun

Title: Classifying z-classes of Weyl groups

Abstract: It is well known that the Weyl group of type A_n is isomorphic to the symmetric group S_{n+1}. The Weyl group of types B_n and C_n are isomorphic to the wreath product of cyclic group C_2 and symmetric group S_n, whereas Weyl group of type D_n is an index 2 subgroup of the Weyl group of C_n. In this talk, we classify the z-classes of these Weyl groups.

Upendra Kulkarni

CMI

Title: Generators-relations and representations for a deformation of FI

Abstract: I will discuss a q-deformation of the rook category and the subcategory of finite injections. These categories are obtained by building on Solomon’s work on the q-rook monoid, which is the Iwahori ring of square matrices over a finite field. I will discuss descriptions of these deformed categories by generators and relations, which are local for the smaller category, giving it a monoidal structure. The representations finite injections have been of interest in the past decade and I will indicate parallel results for the q-analogue. I will also briefly indicate many ways in which monoidal diagrammatic categories arise in combinatorics, representation theory and topology.

Abhishek Das

IIT Kanpur

Title: Generalized Casimir operators for Loop Lie superalgebras

Abstract: Let $\mathfrak g$ be the queer Lie superalgebra $\mathfrak q(n)$ over the field of complex numbers $\mathbb C$. For any associative, commutative, and finitely generated $\mathbb C$-algebra $A$ with identity, let $\mathfrak g \otimes A$ be the corresponding Loop Lie superalgebra. In this poster, we define a class of central operators for $\mathfrak g \otimes A$, called the generalized Casimir operators that generalizes the classical Gelfand invariants. We show that they generate the algebra $U(\mathfrak g \otimes A)^{\mathfrak g}$. We also show that the only non-zero $\mathfrak g$-invariants of $U(\mathfrak g \otimes A)$ for $\mathfrak g=\mathfrak p(n)$, the periplectic Lie superalgebra are the scalars.

Archita Gupta

IIT Kanpur

Title: GELFAND PAIRS AND GELFAND MODULES OF GL2(oℓ)

Abstract: Let F be a non-Archimedean local field with ring of integers o, maximal ideal P, and finite residual field F_q, where q is a power of an odd prime p. Let l ≥ 2 be an integer. Define oℓ = o/P^l as the finite local principal ideal ring. Let G = GL2(oℓ) be the group of invertible 2 × 2 matrices over oℓ , and let B be the standard Borel subgroup of G, consisting of upper triangular matrices. In this poster, we demonstrate that the pair (G, B) forms a strong Gelfand pair. Additionally, we explore the structure and properties of the degenerate Gelfand-Graev modules for G and investigate their applications in the tensor product problem for the regular representations for this group.

Deep H. Makadiya

IIT Bombay

Title: Normal subgroups of twisted Chevalley groups

Abstract: This presentation focuses on the classification of subgroups of twisted Chevalley groups (over a commutative ring) that are normalized by their elementary subgroups. As a significant corollary, we provide a characterization of all normal subgroups of elementary twisted Chevalley groups. This research was conducted in collaboration with Shripad M. Garge.

GV Krishna Teja

ISI Bangalore

TBA

Km Dimpi

IISER Thiruvananthapuram

Title: Hook fusion Procedure for hyperoctahedral groups

Abstract: The fusion procedure was first introduced in the context of constructing new solutions to the Yang-Baxter equations. From a representation-theoretic perspective, it provides a method to construct the primitive idempotents of the group algebra C(S_n). Specifically, a complete set of primitive idempotents, indexed by standard Young tableaux, can be obtained by evaluating certain rational functions in several variables at specific limits. James Grime introduced a version of this process, known as the hook fusion procedure, which reduces the number of auxiliary parameters involved. In this poster presentation, we will explore analogues of the hook fusion procedure for symmetric groups extended to hyperoctahedral groups. The discussion will focus on adapting the procedure and examining its implications in this broader context.

Niranjan Nehra

M.K. Government Polytechnic Jattal, Panipat

Title: Graded character of fusion modules in classical Lie algebras

Abstract: For any finite-dimensional simple Lie algebra $\mathfrak{g}$ of types $A, B, C,$ or $D$. We determine the graded decompositions of fusion products of finite- dimensional irreducible representations corresponding to dominant weights $\lambda$ and $\mu$, both multiples of either the first or the last fundamental weight. Additionally, we provided the graded character for this fusion module.

Papi Ray

IIT Kanpur

Title: A Relationship Between Character Values Of Wreath Products And The Symmetric Group

Abstract: A relation between certain irreducible character values of the hyperoctahedral group Bn (Z/2Z ≀ Sn) and the symmetric group S2n was proved by F. L¨ubeck and D. Prasad in 2021, using Lie theory. In 2022, R. Adin and Y. Roichman proved a similar relation between certain character values of G ≀ Sn and Srn, where G is an abelian group of order r (generalizing the result of L¨ubeck-Prasad) using combinatorial methods. We prove yet another relation between certain irreducible character values of G ≀ Sn and Srn, where G is an abelian group of order r.

Prachi Saini

IISER Pune

Title: Images of Polynomial Maps with Constants

Abstract: Inspired by the Kaplansky-L\'{v}ov conjecture, we investigate the images of polynomials with constants in the free algebra \(\M(2, K)\langle x, y\rangle\), where \(\M(2, K)\) is the \(2 \times 2\) matrix algebra over an algebraically closed field \(K\). We focus on two polynomial maps: (a) the generalized sum of powers \(Ax^{k_1} + By^{k_2}\) and (b) the generalized commutator map \(Axy - Byx\), with \(A\) and \(B\) as non-zero elements of \(\M(2, K)\). Our results show that the images of these maps form vector spaces. It is a collaborative work with Prof. Anupam Singh and Dr. Saikat Panja.

Rijubrata Kundu

IMSc Chennai

Title: Covering Numbers of Some Irreducible Characters of the Symmetric Group

Abstract: The covering number of a non-linear character $\chi$ of a finite group $G$ is the least positive integer $k$ (if it exists) such that every irreducible character of $G$ occurs in $\chi^k$. In 1986, Arad and Herzog showed that covering number exists for every non-linear irreducible character of $G$, whenever $G$ is a finite non-abelian simple group. The character covering number of a finite group $G$, denoted by $\text{ccn}(G)$, is the least positive integer $k$ such that every irreducible character of $G$ occurs in $\chi^k$ for any non-linear irreducible character $\chi$ of $G$. The result of Arad and Herzog shows that $\text{ccn}(G)$ exists whenever $G$ is a finite non-abelian simple group. In a recent work, Miller has shown $\ccn(S_n)$ exists for $n\geq 5$ and that $\text{ccn}(S_n)=n-1$. Here, $S_n$ is the symmetric group on $n$ letters. In this poster presentation, we will display some results in this direction, in particular focusing on the symmetric group. In particular, we will mention the covering number of those irreducible characters of the symmetric group that are indexed by certain two-row partitions or certain hook partitions (this is a joint work with Velmurugan S). This problem for the symmetric group is naturally related to the well-known Kronecker problem, which is a major open problem in the representation theory of the symmetric groups.

Sadhanandh Vishwanath

CMI

Title: qFI category and its representations

Abstract: The representation theory of the FI category, comprising finite sets and injections, has garnered significant attention for its role in studying representation stability and polynomial growth in S_n representations. This framework enables the treatment of compatible families of symmetric group representations as a unified object. Motivated by this, we aimed to develop a q -analogue, the q -FI category, in which the symmetric groups are replaced by the Iwahori–Hecke algebras of type A. This allows us to explore the similarities further between the representation theories of S_n and its q -analogue. Our q-FI draws inspiration from Solomon’s work on q -rook monoids, which uses the Bruhat decomposition for GL_n and its associated reductive monoid M_n . It is also a solution to the Bergman–Elias style problem, leading to a natural monoidal structure that serves as a deformation of the monoidal structure on FI. Furthermore, it has connections with recent work by Geetha and Dipper on Hecke action on tensor powers of natural representations.

Shushma Rani

IISc Bangalore

Title: Truncated Weyl modules for type $A_2$

Abstract: Truncated Weyl modules are the local Weyl modules for the truncated current algebras. We will show that the special case of truncated Weyl modules for type $A_2$ is isomorphic to fusion product modules, hence it gives independence of these fusion product modules from fusion parameters (proof of special case of Feigin Loktev conjecture). We will also see the graded character of these truncated Weyl modules. These truncated Weyl modules helps to give filtration of tensor product of local Weyl modules, $W_{loc}(k_1\omega_1) \otimes W_{loc}(k_2 \omega_2)$ where $\omega_1, \omega_2$ are fundamental weights of Lie algebra $A_2$.

Sudipta Mukherjee

IISER Mohali

Title: Integrable modules for loop affine-Virasoro algebras

Abstract: Affine Kac-Moody Lie algebras and Virasoro algebras are very fundamental in both Mathematics and Physics. The Virasoro algebra acts on the derived algebra of an affine Kac-Moody algebra by derivations. The emerging semidirect product called the Affine-Virasoro algebra turns out to be an important Lie algebra of study as it has applications to conformal field theory, number theory and soliton theory. On the other hand loop algebras are also studied extensively for simple Lie algebras, affine Kac-Moody Lie algebras and Virasoro algebras. In this article, we classify the irreducible integrable modules for the loop affine-Virasoro algebra.

Sulakhana Chowdhury

IISER Thiruvananthapuram

Title: Comparing cohomology via exact split pairs in diagram algebras

Abstract: In this poster, we study those diagram algebras that are cellular (can be written as an iterated inflation of smaller cellular algebras). Our main result establishes a sufficient condition for exact split pairs between the categories of diagram algebras and the categories of its smaller algebras, analogous to a work by Diracca and Koenig. This result allows us to compare the cohomology between these two categories. To be precise, we show the existence of the exact split pairs in $A$-Brauer algebras, cyclotomic Brauer algebras, and walled Brauer algebras with their respective input algebras. This is a joint work with Geetha Thangavelu

TV Ratheesh

IMSc Chennai

TBA

V. Sathish Kumar

HRI

Title: A bijection between two branching models

Abstract: We prove a bijection between the branching models of Kwon and Sundaram, conjectured by Lenart--Lecouvey. To do som we use a symmetry of the Littlewood-Richardson coefficients in terms of the hive model. Along the way, we introduce a new branching rule with flagged hives.

Velmurugan S

IMSc

Title: On minimal polynomial of elements in symmetric and alternating groups

Abstract: Let $ (\rho, V) $ be an irreducible representation of the symmetric group $ S_n$ (or the alternating group $ A_n$), and let $ g $ be a permutation on $n$ letters with each of its cycle lengths divides the length of its largest cycle. We describe completely the minimal polynomial of $\rho(g)$, showing that, in most cases, it equals $x^{o(g)} - 1 $, with a few explicit exceptions. As a by-product, we obtain a new proof (using only combinatorics and representation theory) of a theorem of Swanson that gives a necessary and sufficient condition for the existence of a standard Young tableau of a given shape and major index $r \ \text{mod} \ n$, for all $r$. Thereby, we give a new proof of a celebrated result of Klyachko on Lie elements in a tensor algebra, and of a conjecture of Sundaram on the existence of an invariant vector for $n$-cycles. We also show that for elements $g$ in $S_n$ or $A_n$ of even order, in most cases, $\rho(g)$ has eigenvalue $-1$, with a few explicit exceptions.