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Publications (61)

alg-geom1995

Infinite Grassmannians and Moduli Spaces of G-bundles

Shrawan Kumar

This lecture note is devoted to prove that the space of vacua is isomorphic with the space of generalized theta functions (joint work with M.S. Narasimhan, and A. Ramanathan)

math.AG2013

The multiplicative eigenvalue problem and deformed quantum cohomology

Prakash Belkale, Shrawan Kumar

We construct deformations of the small quantum cohomology rings of homogeneous spaces G/P, and obtain an irredundant set of inequalities determining the multiplicative eigenvalue p…

math.AG2004

On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

Shrawan Kumar

We give a uniform proof of a significant part of the Cachazo-Douglas-Seiberg-Witten conjecture by using topology and geometry of infinite Grassmannians.

math.KT2025

Conjectural Positivity for Pontryagin Product in Equivariant K-theory of Loop Groups

Shrawan Kumar

Let $G$ be a connected simply-connected simple algebraic group over $C$ and let $T$ be a maximal torus, $B \supset T$ a Borel subgroup and $K$ a maximal compact subgroup. T…

math.GR2011

An algebro-geometric realization of equivariant cohomology of some Springer fibers

Shrawan Kumar, Claudio Procesi

We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as an algebra with the action of the Weyl group) with the equivariant cohomology of some Springer…

math.AG2008

On positivity in T-equivariant K-theory of flag varieties

William Graham, Shrawan Kumar

We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a…

math.AG2004

Eigenvalue problem and a new product in cohomology of flag varieties

Prakash Belkale, Shrawan Kumar

Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of…

math.RT2021

Root components for tensor product of affine Kac-Moody Lie algebra modules

Samuel Jeralds, Shrawan Kumar

Let g be an affine Kac-Moody Lie algebra and let $I^», I^¼$ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root $I^^{2}$ , $V(Î»)\otim…

math.GR2015

Components of V(Ï)\otimes V(Ï)

Shrawan Kumar

Kostant asked the following question: Let $g$ be a simple Lie algebra over the complex numbers. Let $I^»$ be a dominant integral weight. Then, $V (I^»)$ is a component of $…

cs.SE2014

Sliced Slices: Separating Data and Control Influences

Shrawan Kumar, Amitabha Sanyal, Uday Khedker

Backward slicing has been used extensively in program understanding, debugging and scaling up of program analysis. For large programs, the size of the conventional backward slice i…

math.RT2008

A generalization of Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces

Shrawan Kumar

We extend the original Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces.

math.RT2021

Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

Shrawan Kumar

We determine the Lie subalgebra $g_{ni l}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $g$ generated by $a d$ -locally nilpotent elements and…

math.AG2006

Explicit Determination of the Picard Group of Moduli Spaces of Semi-Stable G-Bundles on Curves

Arzu Boysal, Shrawan Kumar

Let $C$ be a smooth irreducible projective curve over the complex numbers and let $G$ be a simple simply-connected complex algebraic group. Let $\mathfrak M=\mathfrak M(G,…

math.AG2022

Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells

Shrawan Kumar

Following some work of Aluffi-Mihalcea-SchÃ¼rmann-Su for the CSM classes of Schubert cells and some elaborate computer calculations by R. Rimanyi and L. Mihalcea, I conjecture that…

math.GR2024

Conformal blocks for Galois covers of algebraic curves

Jiuzu Hong, Shrawan Kumar

We study the spaces of twisted conformal blocks attached to a $I^$ -curve $I^£$ with marked $I^$ -orbits and an action of $I^$ on a simple Lie algebra $g$ , where $I^$ is a…

math.AG2009

Finiteness of the number of compatibly-split subvarieties

Shrawan Kumar, Vikram B. Mehta

We show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split.

math.RT2013

Dimension of zero weight space: An algebro-geometric approach

Shrawan Kumar, Dipendra Prasad

Let G be a connected, adjoint, simple algebraic group over the complex numbers with a maximal torus T and a Borel subgroup B containing T. The study of zero weight spaces in irredu…

math.AG2007

Descent of line bundles to GIT quotients of flag varieties by maximal torus

Shrawan Kumar

Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line…

math.RT2007

Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups

Prakash Belkale, Shrawan Kumar

We consider the eigenvalue problem and the associated intersection theory of homogenous spaces for the symplectic and odd orthogonal groups. We solve the Horn and saturation proble…

math.KT2016

Positivity in T-Equivariant K-theory of flag varieties associated to Kac-Moody groups

Shrawan Kumar

We prove a positivity result for the T-equivariant K-theory of flag varieties associated to any symmetrizable Kac-Moody group.

math.RT2013

Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Nathaniel Bushek, Shrawan Kumar

Let $\g$ be any simple Lie algebra over $C$ . Recall that there exists an embedding of $sl_{2}$ into $\g$, called a principal TDS, passing through a principal nilp…

cs.CV2022

Implementing a Real-Time, YOLOv5 based Social Distancing Measuring System for Covid-19

Narayana Darapaneni, Shrawan Kumar, Selvarangan Krishnan +4

The purpose of this work is, to provide a YOLOv5 deep learning-based social distance monitoring framework using an overhead view perspective. In addition, we have developed a custo…

math.RT2024

Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula

Shrawan Kumar, Dipendra Prasad

Let $G$ be a connected reductive group over the complex numbers and let $T \subset G$ be a maximal torus. For any $t \in T$ of finite order and any irreducible representation $V (I^»)$ …

cs.SE2022

Automated Validation of Insurance Applications against Calculation Specifications

Advaita Datar, Amey Zare, Asia A +3

Insurance companies rely on their Legacy Insurance System (LIS) to govern day-to-day operations. These LIS operate as per the companys business rules that are formally specified in…

math.SG2003

The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type

Shrawan Kumar, Bernhard Leeb, John J. Millson

We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces. We find that for Sp(6) the corresponding polyhedral cone has 102 facets and 51 edges.

math.KT2017

Positivity in $T$ -equivariant $K$ -theory of flag varieties associated to Kac-Moody groups II

Seth Baldwin, Shrawan Kumar

We prove sign-alternation of the structure constants in the basis of structure sheaves of opposite Schubert varieties in the torus-equivariant Grothendieck group of coherent sheave…

math.GR2008

A conjectural presentation of fusion algebras

Arzu Boysal, Shrawan Kumar

Let g be a semisimple Lie algebra over the complex numbers. Fix a positive integer l (called the level). Let R(l,g) be the fusion algebra at level l. Then, there is an algebra homo…

cs.PL2015

Property irrelevant predicates

Shrawan Kumar

Although slicing removes code which has no bearing on property checking. However even after that, our study has found that there are predicates in program which have no bearing on…

math.RT2013

A study of saturated tensor cone for symmetrizable Kac-Moody algebras

Merrick Brown, Shrawan Kumar

Let $\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\fh$ and the Weyl group $W$ . Let $P_{+}$ be the set of dominant integral weights. For $Î»\in P…

math.AG2012

An approach towards the KollÃ¡r-Peskine problem via the Instanton Moduli Space

Shrawan Kumar

We look at the following question raised by KollÃ¡r and Peskine. (Actually, it is a slightly weaker version of their question.) Let $V_{t}$ be a family of rank two vector bundles on…

math.RT2007

Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups

Shrawan Kumar, George Lusztig, Dipendra Prasad

In this paper we show how a theorem of Jantzen relating the character of a finite dimensional irreducible representation of a disconnected semisimple algebraic group on elements ou…

math.AG2023

Nonexistence of regular maps between homogeneous projective varieties

Shrawan Kumar

We study non-existence of non-constant regular maps from a partial flag variety $X = G / P$ to another partial flag variety $X^{'} = G^{'} / P^{'}$ and prove that there does not exist any non-const…

math.AG2002

A new realization of the cohomology of Springer fibers

Shrawan Kumar, Jesper Funch Thomsen

In this note we give a geometric realization of the cohomology of Springer fibers in type A. More precisely, we describe the cohomology by the coordinate ring of a scheme theoretic…

math.RT2015

Representation ring of Levi subgroups versus cohomology ring of flag varieties

Shrawan Kumar

Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Little…

alg-geom1995

The nil Hecke ring and singularity of Schubert varieties

Shrawan Kumar

We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.

math.AG2019

Elliptic classes of Schubert varieties

Shrawan Kumar, RichÃ¡rd RimÃ¡nyi, Andrzej Weber

We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not d…

alg-geom1995

Picard group of the moduli spaces of G-bundles

Shrawan Kumar, M. S. Narasimhan

We prove that the Picard group of the moduli space of semistable G-bundles on any irreducible smooth projective curve is isomorphic with the group of integers.

math.AG1999

Frobenius splitting of Hilbert schemes of points on surfaces

Shrawan Kumar, Jesper Funch Thomsen

Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. We show that if X is Frobenius split then so is the Hilbert scheme…

math.AG2003

Equivariant analogue of Grothendieck's theorem for vector bundles on $P^{1}$

Shrawan Kumar

We prove an equivariant analogue of Grothendieck's theorem for vector bundles on the one dimensional projective space over complex numbers.

math.RT2022

Category $C_{k}$ of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

Shrawan Kumar

Lusztig defined an abelian category $C_{k}$ of a class of representations of a multi-loop algebra and asked various questions connecting it to the modular representatio…

math.AG2010

A generalization of Fulton's conjecture for arbitrary groups

Prakash Belkale, Shrawan Kumar, Nicolas Ressayre

We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.

math.RT2023

Components of $V (I ¨ ) \otimes V (I ¨ )$ and dominant weight polyhedra for affine Kac-Moody Lie algebras

Sam Jeralds, Shrawan Kumar

Kostant asked the following question: Let $g$ be a simple Lie algebra over the complex numbers. Let $I^»$ be a dominant integral weight. Then, $V (I^»)$ is a component of $…

math.RT2019

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

Shrawan Kumar, Sean Rogers

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ Î¾^P_Î»: Rep^\mathbb{C}_{Î»-poly}(L) \to H…

math.AG2002

A conjectural generalization of n! result to arbitrary groups

Shrawan Kumar, Jesper Funch Thomsen

We relate the n! conjecture (by Garsia and Haiman) to the geometry of principal nilpotent pairs, and state a conjecture generalizing the n! conjecture to arbitrary semisimple algeb…

math.AG2025

Positivity in $T$ -Equivariant $K$ -theory of partial flag varieties associated to Kac-Moody groups

Joseph Compton, Shrawan Kumar

We prove sign-alternation of the product structure constants in the basis dual to the basis consisting of the structure sheaves of Schubert varieties in the torus-equivariant Groth…

math.QA2004

Algebraization of Frobenius splitting via quantum groups

Shrawan Kumar, Peter Littelmann

An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties…

math.AG2024

On the faces of the tensor cone of symmetrizable Kac-Moody lie algebras

Shrawan Kumar, Nicolas Ressayre

In this paper, we are interested in the decomposition of the tensor product of two representations ofa symmetrizable Kac-Moody Lie algebra $g$ , or more precisely in the…

math.RT2006

Saturation and Irredundancy for Spin(8)

Michael Kapovich, Shrawan Kumar, John J. Millson

We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the sid…

math.AG2010

Geometry of orbits of permanents and determinants

Shrawan Kumar

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.

math.RT2016

A complete set of intertwiners for arbitrary tensor product representations via current algebras

Shrawan Kumar

Let $g$ be a reductive Lie algebra and let $V (I^»)$ be a tensor product of $k$ copies of finite dimensional irreducible $g$ -modules. Choosing $k$ poi…

math.AG2012

Richardson Varieties Have Kawamata Log Terminal Singularities

Shrawan Kumar, Karl Schwede

Let $X_{w}^{v}$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$ . Recall that $X_{w}^{v}$ is the intersection of the finite dimensiona…

math.CO2024

Counter Example to a Strong Matroid Minor Conjecture

Shrawan Kumar

The main result of this note asserts that a strong form of the Matroid Minor Conjecture due to J. Draisma is not true, i.e., there exist properly ascending chains of $S_{\infty}$ -sta…

math.RT2003

Induction Functor in Non-commutative Equivariant Cohomology and Dirac Cohomology

Shrawan Kumar

The aim of this paper is to put some recent results of Huang-Pandzic (conjectured by Vogan) and Kostant on Dirac cohomology in a broader perspective.This is achieved by introducing…

math.AG2013

Additive Eigenvalue Problem (a survey), (With appendix by M. Kapovich)

Shrawan Kumar

The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A+B of two Hermitian matrices A and B, provided we fix the…

math.RT2022

Representation ring of Levi subgroups versus cohomology ring of flag varieties III

Shrawan Kumar, Jiale Xie

For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$ , the first author [Ku2] introduced a ring homomorphism $ Î¾^P_Î»: Rep^\mathbb{C}_{Î»-poly}(L) \t…

math.RT2012

A Study of the representations supported by the orbit closure of the determinant

Shrawan Kumar

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the ce…

math.AG1998

Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

Shrawan Kumar, Niels Lauritzen, Jesper Funch Thomsen

We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only…

math.RT2023

Lie algebra cohomology of the positive part of twisted affine Lie algebras

Jiuzu Hong, Shrawan Kumar

The explicit Verlinde formula for the dimension of conformal blocks, attached to a marked projective curve $I^£$ , a simple Lie algebra $g$ over $C$ and integrabl…

math.RT2007

Special isogenies and tensor product multiplicities

Shrawan Kumar, John R. Stembridge

We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the co…

math.RT2022

Twisted conformal blocks and their dimension

Jiuzu Hong, Shrawan Kumar

Let $I^$ be a finite group acting on a simple Lie algebra $g$ and acting on a $s$ -pointed projective curve $(I^£, p = {p_{1}, \dots, p_{s}})$ faithfully (for $s \geq 1$ )…

math.AG2014

Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

Shrawan Kumar, J. M. Landsberg

We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some in…

Publications (61)

Infinite Grassmannians and Moduli Spaces of G-bundles

The multiplicative eigenvalue problem and deformed quantum cohomology

On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

Conjectural Positivity for Pontryagin Product in Equivariant K-theory of Loop Groups

An algebro-geometric realization of equivariant cohomology of some Springer fibers

On positivity in T-equivariant K-theory of flag varieties

Eigenvalue problem and a new product in cohomology of flag varieties

Root components for tensor product of affine Kac-Moody Lie algebra modules

Components of V(Ï)\otimes V(Ï)

Sliced Slices: Separating Data and Control Influences

A generalization of Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces

Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

Explicit Determination of the Picard Group of Moduli Spaces of Semi-Stable G-Bundles on Curves

Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells

Conformal blocks for Galois covers of algebraic curves

Finiteness of the number of compatibly-split subvarieties

Dimension of zero weight space: An algebro-geometric approach

Descent of line bundles to GIT quotients of flag varieties by maximal torus

Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups

Positivity in T-Equivariant K-theory of flag varieties associated to Kac-Moody groups

Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Implementing a Real-Time, YOLOv5 based Social Distancing Measuring System for Covid-19

Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula

Automated Validation of Insurance Applications against Calculation Specifications

The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type

Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups II

A conjectural presentation of fusion algebras

Property irrelevant predicates

A study of saturated tensor cone for symmetrizable Kac-Moody algebras

An approach towards the KollÃ¡r-Peskine problem via the Instanton Moduli Space

Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups

Nonexistence of regular maps between homogeneous projective varieties

A new realization of the cohomology of Springer fibers

Representation ring of Levi subgroups versus cohomology ring of flag varieties

The nil Hecke ring and singularity of Schubert varieties

Elliptic classes of Schubert varieties

Picard group of the moduli spaces of G-bundles

Frobenius splitting of Hilbert schemes of points on surfaces

Equivariant analogue of Grothendieck's theorem for vector bundles on P1

Category Ck​ of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

A generalization of Fulton's conjecture for arbitrary groups

Components of V(I¨)⊗V(I¨) and dominant weight polyhedra for affine Kac-Moody Lie algebras

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

A conjectural generalization of n! result to arbitrary groups

Positivity in T-Equivariant K-theory of partial flag varieties associated to Kac-Moody groups

Algebraization of Frobenius splitting via quantum groups

On the faces of the tensor cone of symmetrizable Kac-Moody lie algebras

Saturation and Irredundancy for Spin(8)

Geometry of orbits of permanents and determinants

A complete set of intertwiners for arbitrary tensor product representations via current algebras

Richardson Varieties Have Kawamata Log Terminal Singularities

Counter Example to a Strong Matroid Minor Conjecture

Induction Functor in Non-commutative Equivariant Cohomology and Dirac Cohomology

Additive Eigenvalue Problem (a survey), (With appendix by M. Kapovich)

Representation ring of Levi subgroups versus cohomology ring of flag varieties III

A Study of the representations supported by the orbit closure of the determinant

Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

Lie algebra cohomology of the positive part of twisted affine Lie algebras

Special isogenies and tensor product multiplicities

Twisted conformal blocks and their dimension

Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

Components of V(Ï)\otimes V(Ï)

Positivity in $T$ -equivariant $K$ -theory of flag varieties associated to Kac-Moody groups II

Equivariant analogue of Grothendieck's theorem for vector bundles on $P^{1}$

Category $C_{k}$ of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

Components of $V (I ¨ ) \otimes V (I ¨ )$ and dominant weight polyhedra for affine Kac-Moody Lie algebras

Positivity in $T$ -Equivariant $K$ -theory of partial flag varieties associated to Kac-Moody groups