scholarly journals Skew Divided Difference Operators and Schubert Polynomials

2007 ◽  
Author(s):  
Anatol N. Kirillov
Keyword(s):  
Divided Difference ◽  
Difference Operators ◽  
Schubert 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.

scholarly journals Polynomials Defined by Tableaux and Linear Recurrences

10.37236/5284 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Per Alexandersson
Keyword(s):  
Divided Difference ◽  
Lattice Points ◽  
Difference Operators ◽  
Linear Recurrences ◽  
Natural Operation ◽  
Littlewood Polynomials ◽  
Strongly Connected ◽  
Key Polynomials ◽  
Demazure Characters ◽  
Demazure Atoms

We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure atoms. The same technique can be applied to Hall-Littlewood polynomials and dual Grothendieck polynomials.The motivation behind this is that such recurrences are strongly connected with other nice properties, such as interpretations in terms of lattice points in polytopes and divided difference operators.

On divided difference operators in function algebras

2008 ◽  
Vol 41 (2) ◽  
Author(s):  
Piotr Multarzyński
Keyword(s):  
Divided Difference ◽  
Difference Operators ◽  
Difference Quotient ◽  
Partial Difference ◽  
Function Algebras ◽  
Definition Of ◽  
Product Rules ◽  
Set Of Points ◽  
Tension Structure

AbstractIn this paper we study divided difference operators of any order acting in function algebras. In the definition of difference quotient operators we use a tension structure defined on the set of points on which depend the functions of the algebras considered. In the paper we mention the oportunity for partial difference quotient operators as well as for some purely algebraic definition of divided difference operators in terms of the suitable Leibniz product rules.

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 ON DIVIDED-DIFFERENCE SEMIGROUPS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 179-191
Author(s):  
D. G. FITZGERALD
Keyword(s):  
Symmetric Group ◽  
Group Ring ◽  
Divided Difference ◽  
Difference Operators ◽  
Bijective Correspondence ◽  
Diagrammatic Representations ◽  
Skew Group Ring

The Newtonian divided-difference operators generate the nil-Coxeter algebra and semigroup. A bijective correspondence between the nil-Coxeter semigroup and the symmetric group is used to provide braid-like diagrams for the former, and corresponding Reidemeister-type moves for the relations. Conditions are given for similar relations to hold in a skew group ring. Interesting extensions of the nil-Coxeter semigroup are described and given diagrammatic representations.

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