scholarly journals Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

2019 ◽  
Vol 517 ◽  
pp. 19-77
Author(s):  
Christoph Bärligea
Keyword(s):  
Coxeter Group ◽  
Divided Difference ◽  
Difference Operators ◽  
Nichols Algebra ◽  
Finite Coxeter Group

scholarly journals Some remarks on the algebraic structure of the finite Coxeter groupF4

1999 ◽  
Vol 22 (1) ◽  
pp. 81-84
Author(s):  
Muhammad A. Albar ◽  
Norah Al-Saleh
Keyword(s):  
Coxeter Group ◽  
Algebraic Structure ◽  
Finite Coxeter Group

We consider in this paper the algebraic structure and some properties of the finite Coxeter groupF4.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

2019 ◽  
Vol 75 (3) ◽  
pp. 541-550
Author(s):  
Emmanuel Bourret ◽  
Zofia Grabowiecka
Keyword(s):  
Coxeter Group ◽  
Dynkin Diagram ◽  
Geometrical Structure ◽  
Three Dimensions ◽  
Icosahedral Group ◽  
Finite Coxeter Group

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a `pancake structure'.

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 Generalized Descent Algebras

2007 ◽  
Vol 50 (4) ◽  
pp. 535-546
Author(s):  
Christophe Hohlweg
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Descent Algebra ◽  
Finite Coxeter Group ◽  
Solomon Descent Algebra ◽  
Descent Algebras

AbstractIf A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

scholarly journals Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

2020 ◽  
Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Combinatorial Properties ◽  
Coxeter Matroid ◽  
Finite Coxeter Group ◽  
Bruhat Intervals

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

scholarly journals Special involutions and bulky parabolic subgroups in finite Coxeter groups

2005 ◽  
Vol 79 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Götz Pfeiffer ◽  
Gerhard Röhrle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Hyperplane Arrangement ◽  
Conjugacy Classes ◽  
Parabolic Subgroups ◽  
Finite Coxeter Group

AbstractThe conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

scholarly journals Leading coefficients and cellular bases of Hecke algebras

2009 ◽  
Vol 52 (3) ◽  
pp. 653-677 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Cellular Structure ◽  
Hecke Algebras ◽  
Point Of View ◽  
Reflection Groups ◽  
Cellular Basis ◽  
Complex Reflection Groups ◽  
Finite Coxeter Group ◽  
New Construction

AbstractLet H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

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.

Sign in / Sign up

Export Citation Format

Share Document