scholarly journals ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

2016 ◽  
Vol 56 (6) ◽  
pp. 440-447
Author(s):  
Lenka Háková ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Special Functions ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
One Dimensional ◽  
Complex Functions ◽  
Group Orbit ◽  
Rank 2 ◽  
Definition Of

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

scholarly journals ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS

2016 ◽  
Vol 56 (3) ◽  
pp. 202 ◽  
Author(s):  
Lenka Háková ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Polynomials ◽  
Special Functions ◽  
Root Systems ◽  
Full Generality ◽  
Cubature Formulas ◽  
Special Cases ◽  
Group Orbit

The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra <em>C</em><sub>2</sub>.

scholarly journals On flag varieties, hyperplane complements and Springer representations of Weyl groups

1990 ◽  
Vol 49 (3) ◽  
pp. 449-485 ◽  
Author(s):  
G. I. Lehrer ◽  
T. Shoji
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Nilpotent Element ◽  
Parabolic Type ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Flag Varieties ◽  
Group Representations ◽  
Definition Of

AbstractLet G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

scholarly journals THE QUANTUM G2 LINK INVARIANT

1994 ◽  
Vol 05 (01) ◽  
pp. 61-85 ◽  
Author(s):  
GREG KUPERBERG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
The Other ◽  
Homfly Polynomial ◽  
Link Invariant ◽  
Simple Lie Algebra ◽  
Inductive Definitions ◽  
Rank 2 ◽  
Kauffman Polynomial ◽  
Definition Of

We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G2. It is therefore related to G2 in the same way that the HOMFLY polynomial is related to An and the Kauffman polynomial is related to Bn, Cn, and Dn. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.

scholarly journals Faces of Platonic solids in all dimensions

2014 ◽  
Vol 70 (4) ◽  
pp. 358-363 ◽  
Author(s):  
Marzena Szajewska
Keyword(s):  
Lie Algebras ◽  
Dynkin Diagram ◽  
Coxeter Groups ◽  
Weyl Groups ◽  
Platonic Solids ◽  
Dual Pairs ◽  
Essential Information ◽  
Specific Dimension ◽  
Group Orbit ◽  
Real Euclidean Space

This paper considers Platonic solids/polytopes in the real Euclidean space {\bb R}^n of dimension 3 ≤n< ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤d≤n− 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of typesAn,Bn,Cn,F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groupsH3,H4. The method consists of recursively decorating the appropriate Coxeter–Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit,i.e.are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.

H-Finite Irreducible Representations of Simple Lie Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1084-1106 ◽  
Author(s):  
F. Lemire ◽  
M. Pap
Keyword(s):  
Irreducible Representation ◽  
Lie Algebras ◽  
Complex Number ◽  
Cartan Subalgebra ◽  
Irreducible Representations ◽  
Weight Space ◽  
Algebra Homomorphism ◽  
One Dimensional ◽  
Enveloping Algebra ◽  
Complex Number Field

Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.

scholarly journals Modular Irreducible Representations of the FpW4-Submodules ,()pFNof the Modules ,()pFMas Linear Codes, where W4is the Weyl Group of Type B4

2021 ◽  
Vol 24 (2) ◽  
pp. 48-63
Author(s):  
Jinan F. N. Al-Jobory ◽  
◽  
Emad B. Al-Zangana ◽  
Faez Hassan Ali ◽  
◽  
...  
Keyword(s):  
Positive Integer ◽  
Weyl Group ◽  
Prime Number ◽  
Linear Codes ◽  
Galois Field ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Modular Representations ◽  
Generator Matrix ◽  
Specht Modules

The modular representations of the FpWn-Specht modules( , )KSas linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules( , )pFNof the Specht modules pFS ( , )as linear codes where W4is the Weyl group of type B4is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules( , )pFNof the FpW4-modules( , )pFMfor each pair of partitions( , )of a positive integer n4, where FpGF(p) is the Galois field (finite field) of order p, and pis a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace((2,1),(1))()pU representing the irreducible FpW4-submodules((2,1),(1))pFNof the FpW4-modules((2,1),(1))pF Mand give the linear code of ((2,1),(1))()pU for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces( , )()pUfor all pair of partitions( , )of a positive integer n4, and for each prime number p greater than or equal to 3.We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn

scholarly journals THE CONSTRUCTION OF TRIGONOMETRIC INVARIANTS FOR WEYL GROUPS AND THE DERIVATION OF THE CORRESPONDING EXACTLY SOLVABLE SUTHERLAND MODELS

1999 ◽  
Vol 14 (14) ◽  
pp. 937-949 ◽  
Author(s):  
O. HASCHKE ◽  
W. RÜHL
Keyword(s):  
Weyl Group ◽  
Polynomial Algebra ◽  
Solvable Model ◽  
Weyl Groups ◽  
Root Lattice ◽  
Root Space ◽  
Exactly Solvable Model ◽  
Exactly Solvable ◽  
Group Orbit

Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the co-root lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the co-root space and lead to an exactly solvable model of Sutherland type. We apply this construction to the F4 case.

On the characters of the Weyl group of type D

1975 ◽  
Vol 77 (2) ◽  
pp. 259-264 ◽  
Author(s):  
S. J. Mayer
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Unified Theory ◽  
Reflection Groups ◽  
Type D ◽  
Common Structure ◽  
Principal Character ◽  
Similar Development

This paper is a continuation of (2), (3) in the development of a unified theory of the characters of the Weyl groups of the simple Lie algebras using their common structure as reflection groups; compare Carter (1) for a similar development for the conjugacy classes. We look at the Weyl group of type D, which is a subgroup of index two in the Weyl group of type C. It was first studied by Young (4), but rather less is known about the characters of this group than those of types A and C. Indeed, the situation is rather more complicated, but we are able to give, as before, an algorithm to determine irreducible constituents of the principal character of a Weyl subgroup induced up to the whole group. We shall also study the case where the rank of the Weyl group is even, when extra irreducible characters may arise, and after constructing these, we shall state some results on their occurrence in the induced principal character.

scholarly journals Characteristic cycles and the microlocal geometry of the Gauss map, II

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Krämer
Keyword(s):  
Weak Solution ◽  
Weyl Group ◽  
Lie Algebras ◽  
Abelian Variety ◽  
Cotangent Bundle ◽  
Abelian Varieties ◽  
Gauss Map ◽  
Schottky Problem ◽  
Group Orbit

Abstract We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

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