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Shrawan Kumar

Publications and source records attributed to Shrawan Kumar.

At least 37 records · Page 2Linked to original sources
arXiv2019

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

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ ξ_{n,k}: Rep^\mathbb{C}_{ω_1-poly}(Sp(2k)) \to H^*(IG(n-k, 2n), \mathbb{C})$. Our main result asserts that $ ξ_{n,k}$ is injective when n tends to $\infty$ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

math.RT↗
arXiv2017

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

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 sheaves on the flag varieties $G/P$ associated to an arbitrary symmetrizable Kac-Moody group $G$, where $P$ is any parabolic subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite case to the general Kac-Moody case, and affirmatively answers a conjecture of Lam-Schilling-Shimozono regarding the signs of the structure constants in the case of the affine Grassmannian.

math.KT↗
arXiv2016

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

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\vec{V}(\vecλ)$ be a tensor product of $k$ copies of finite dimensional irreducible $\mathfrak{g}$-modules. Choosing $k$ points in $\mathbb{C}$, $\vec{V}(\vecλ)$ acquires a natural structure of the current algebra $\mathfrak{g}\otimes \mathbb{C}[t]$-module. Following a work of Rao [R], we produce an explicit and complete set of $\mathfrak{g}$-module intertwiners of $\vec{V}(\vecλ)$ in terms of the action of the current algebra.

math.RT↗
arXiv2015

Representation ring of Levi subgroups versus cohomology ring of flag varieties

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 Littlewood-Richardson tensor product structure constants for GL(r). Specifically, the result asserts that there is an explicit ring homomorphism ϕ: \Rep _{\poly}(GL(r)) to H^*(Gr(r, n)), where Gr(r, n) denotes the Grassmannian of r-planes in C^n and \Rep_{\poly} (GL(r)) denotes the polynomial representation ring of GL(r). This work seeks to achieve one possible generalization of this classical result for GL(r) and the Grassmannian Gr(r,n) to the Levi subgroups of any reductive group G and the corresponding flag varieties.

math.RT↗
arXiv2015

Components of V(ρ)\otimes V(ρ)

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $λ$ be a dominant integral weight. Then, $V(λ)$ is a component of $ V(ρ)\otimes V(ρ)$ if and only if $λ\leq 2ρ$ under the usual Bruhat-Chevalley order on the set of weights. We give an affirmative answer to this question up to a saturation factor. In particular, the question is answered affirmatively for the special linear Lie algebras due to the Saturation Theorem of Knutson-Tao.

math.GR↗
arXiv2015

Property irrelevant predicates

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 property validation, although slicing could not eliminate them. We have cope up with a criteria to identify such predicates and then give a process to leverage them in scale up of property checking.

cs.PL↗
arXiv2014

Sliced Slices: Separating Data and Control Influences

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 is about 25% of the program size. This may be too large to be useful. Our investigations reveal that in general, the size of a slice is influenced more by computations governing the control flow reaching the slicing criterion than by the computations governing the values relevant to the slicing criterion. We distinguish between the two by defining data slices and control slices both of which are smaller than the conventional slices which can be obtained by combining the two. This is useful because for many applications, the individual data or control slices are sufficient. Our experiments show that for more than 50% of cases, the data slice is smaller than 10% of the program in size. Besides, the time to compute data or control slice is comparable to that for computing the conventional slice.

cs.SE↗
arXiv2013

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

Let $\g$ be any simple Lie algebra over $\mathbb{C}$. Recall that there exists an embedding of $\mathfrak{sl}_2$ into $\g$, called a principal TDS, passing through a principal nilpotent element of $\g$ and uniquely determined up to conjugation. Moreover, $\wedge (\g^*)^\g$ is freely generated (in the super-graded sense) by primitive elements $ω_1, \dots, ω_\ell$, where $\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive element $ω\in \wedge^d (\g^*)^\g$, there exists an irreducible $\mathfrak{sl}_2$-submodule $V_ω\subset \g$ of dimension $d$ such that $ω$ is non-zero on the line $\wedge^d (V_ω)$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let G be a connected, simply-connected, simple, simply-laced algebraic group and let $σ$ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G) \to R(K) is surjective, where R denotes the representation ring over $\mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology H^*(G)\to H^*(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.

math.RT↗
arXiv2013

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

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 $λ\in P_+$, let $L(λ)$ be the irreducible, integrable, highest weight representation of $\fg$ with highest weight $λ$. For a positive integer $s$, define the {\em saturated tensor semigroup} as \begin{align*} Γ_s:= \{(λ_1, \dots, λ_s,μ)\in P_+^{s+1}: \exists\, N>1 \,\,\text{with}\,\, L(Nμ)\subset L(Nλ_1)\otimes \dots \otimes L(Nλ_s)\}. \end{align*} The aim of this paper is to begin a systematic study of $Γ_s$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $Γ_s$. We further prove that any integer $d>0$ is a saturation factor for $A^{(1)}_1$ and 4 is a saturation factor for $A^{(2)}_2$.

math.RT↗
arXiv2013

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

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 eigenvalues of A and B. A systematic study of this problem was initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians, notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson-Tao (1999), the problem is finally settled. The solution asserts that the eigenvalues of A+B are given in terms of certain system of linear inequalities in the eigenvalues of A and B. These inequalities are given explicitly in terms of certain triples of Schubert classes in the singular cohomology of Grassmannians and the standard cup product. Belkale (2001) gave an optimal set of inequalities for the problem in this case. The Hermitian eigenvalue problem has been extended by Berenstein-Sjamaar (2000) and Kapovich-Leeb-Millson (2005) for any semisimple complex algebraic group G. Their solution is again in terms of a system of linear inequalities obtained from certain triples of Schubert classes in the singular cohomology of the partial flag varieties G/P (P being a maximal parabolic subgroup) and the standard cup product. However, their solution is far from being optimal. In a joint work with P. Belkale, we define a deformation of the cup product in the cohomology of G/P and use this new product to generate our system of inequalities which solves the problem for any G optimally (as shown by Ressayre). This article is a survey (with more or less complete proofs) of this additive eigenvalue problem.

math.AG↗
arXiv2013

Dimension of zero weight space: An algebro-geometric approach

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 irreducible representations of G has been a topic of considerable interest; there are many works which study the zero weight space as a representation space for the Weyl group. In this paper, we study the variation on the dimension of the zero weight space as the irreducible representation varies over the set of dominant integral weights for T which are lattice points in a certain polyhedral cone. The theorem proved here asserts that the zero weight spaces have dimensions which are piecewise polynomial functions on the polyhedral cone of dominant integral weights.

math.RT↗
arXiv2012

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

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 celebrated `Latin Square Conjecture' due to Alon-Tarsi or more precisely on the validity of an equivalent `column Latin Square Conjecture' due to Huang-Rota.

math.RT↗
arXiv2012

Richardson Varieties Have Kawamata Log Terminal Singularities

Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $Δ$ on $X^v_w$ and prove that the pair $(X^v_w, Δ)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - Δ$ is ample, which additionally proves that $(X^v_w, Δ)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities.

math.AG↗
arXiv2012

An approach towards the Kollár-Peskine problem via the Instanton Moduli Space

We look at the following question raised by Kollár and Peskine. (Actually, it is a slightly weaker version of their question.) Let $V_t$ be a family of rank two vector bundles on $\Bbb P^3$. Assume that the general member of the family is a trivial vector bundle. Then, is the special member $V_0$ also a trivial vector bundle? We show that this question is equivalent to the nonexistence of morphisms from $\Bbb P^3\to \mathcal{X}$, where $\mathcal{X}$ is the infinite Grassmannian associated to SL(2). We further reduce this question to the nonexistence of $\Bbb C^*$-equivariant morphisms from $\Bbb C^3\setminus \{0\} \to \mathcal{M}_d$ (for any $d>0$), where $\mathcal{M}_d$ is the Donaldson moduli space of isomorphism classes of rank two vector bundles $\mathcal{V}$ over $\Bbb P^2$ with trivial determinant and with second Chern class $d$ together with a trivialization of $\mathcal{V}_{|\Bbb P^1}$.

math.AG↗