scholarly journals Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

2021 ◽  
Author(s):  
Leonardo C Mihalcea ◽  
Hiroshi Naruse ◽  
Changjian Su
Keyword(s):  
Divided Difference ◽  
Quantum Cohomology ◽  
Flag Manifold ◽  
Leibniz Rule ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Left Multiplication ◽  
Schubert Classes ◽  
Left And Right ◽  
K Theory

Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

scholarly journals Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

2016 ◽  
Vol 152 (12) ◽  
pp. 2603-2625 ◽  
Author(s):  
Paolo Aluffi ◽  
Leonardo C. Mihalcea
Keyword(s):  
Divided Difference ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Generalized Flag Manifold ◽  
Effective Combination ◽  
A Cell ◽  
Schubert Classes ◽  
Dimension One ◽  
Schubert Class

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

scholarly journals Quantum K-Theory of Grassmannians and Non-Abelian Localization

2021 ◽  
Author(s):  
Alexander Givental ◽  
◽  
Xiaohan Yan ◽  
Keyword(s):  
Finite Difference ◽  
Hypergeometric Series ◽  
Rational Curves ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Jackson Type ◽  
Quiver Varieties ◽  
K Theory

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

scholarly journals Wilson loop algebras and quantum K-theory for Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hans Jockers ◽  
Peter Mayr ◽  
Urmi Ninad ◽  
Alexander Tabler
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Wilson Line ◽  
Quantum Cohomology ◽  
Three Dimensional ◽  
Loop Algebra ◽  
Loop Algebras ◽  
Verlinde Algebra ◽  
K Theory ◽  
Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

scholarly journals Equivariant $K$ -theory of affine flag manifolds and affine Grothendieck polynomials

2009 ◽  
Vol 148 (3) ◽  
pp. 501-538 ◽  
Author(s):  
Masaki Kashiwara ◽  
Mark Shimozono
Keyword(s):  
Flag Manifolds ◽  
K Theory ◽  
Affine Flag ◽  
Grothendieck Polynomials

scholarly journals Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1251
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Sonia Bhalla ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Divided Difference ◽  
Difference Operators ◽  
New Variant ◽  
Order Four ◽  
With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

scholarly journals GLSMs for exotic Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Wei Gu ◽  
Eric Sharpe ◽  
Hao Zou
Keyword(s):  
Global Symmetries ◽  
Sigma Models ◽  
Quantum Cohomology ◽  
Flag Manifolds ◽  
Coulomb Branch ◽  
Linear Sigma Models ◽  
Symplectic Grassmannians ◽  
Orthogonal Grassmannians

Abstract In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.

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