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.

scholarly journals Non-symmetric Cauchy kernels

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Olga Azenhas ◽  
Aram Emami
Keyword(s):  
Type A ◽  
Cauchy Kernel ◽  
Double Crystal ◽  
Crystal Graphs ◽  
Sum Of Products ◽  
Key Polynomials ◽  
Demazure Characters ◽  
Rsk Algorithm ◽  
International Audience ◽  
Demazure Atoms

International audience Using an analogue of the Robinson-Schensted-Knuth (RSK) algorithm for semi-skyline augmented fillings, due to Sarah Mason, we exhibit expansions of non-symmetric Cauchy kernels $∏_(i,j)∈\eta (1-x_iy_j)^-1$, where the product is over all cell-coordinates $(i,j)$ of the stair-type partition shape $\eta$ , consisting of the cells in a NW-SE diagonal of a rectangle diagram and below it, containing the biggest stair shape. In the spirit of the classical Cauchy kernel expansion for rectangle shapes, this RSK variation provides an interpretation of the kernel for stair-type shapes as a family of pairs of semi-skyline augmented fillings whose key tableaux, determined by their shapes, lead to expansions as a sum of products of two families of key polynomials, the basis of Demazure characters of type A, and the Demazure atoms. A previous expansion of the Cauchy kernel in type A, for the stair shape was given by Alain Lascoux, based on the structure of double crystal graphs, and by Amy M. Fu and Alain Lascoux, relying on Demazure operators, which was also used to recover expansions for Ferrers shapes.

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.

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 Lattice Points in the Newton Polytopes of Key Polynomials

2020 ◽  
Vol 34 (2) ◽  
pp. 1281-1289
Author(s):  
Neil J. Y. Fan ◽  
Peter L. Guo ◽  
Simon C. Y. Peng ◽  
Sophie C. C. Sun
Keyword(s):  
Lattice Points ◽  
Newton Polytopes ◽  
Key Polynomials

scholarly journals A $q$-Analog of Foulkes' Conjecture

10.37236/6004 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
François Bergeron
Keyword(s):  
Symmetric Functions ◽  
Divided Difference ◽  
Supporting Evidence ◽  
Littlewood Polynomials

We propose a $q$-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials $H_n(\boldsymbol{x};q)$ replaces the analogous difference of plethysms of complete homogeneous symmetric functions $h_n(\boldsymbol{x})$ in Foulkes' conjecture. At $q=0$, we get back the original statement of Foulkes, and we show that our version holds at $q=1$. We discuss further supporting evidence, as well as various generalizations, including a $(q,t)$-version.

Sign in / Sign up

Export Citation Format

Share Document