Marzena Szajewska

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All published works (23)

Construction of Low-Dimensional Lie Algebras versus Eigenproblem

2025-05-02

Alina Dobrogowska , Marzena Szajewska

Eigenvalue problem versus Casimir functions for Lie algebras

2024-03-25

Alina Dobrogowska , Marzena Szajewska

On Some Structures of Lie Algebroids on the Cotangent Bundles

2023-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

2020-10-20

Mariia Myronova , J. Patera , Marzena Szajewska

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and … The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of $k$ orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups $H_2$ and $H_3$. The geometrical structures of nested polytopes are exemplified.

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Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

2019-06-03

Jiří Hrivnák , J. Patera , Marzena Szajewska

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality … We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified.

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Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska +1 more

The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras. The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras.

The orthogonal systems of functions on lattices of SU(n + 1), n < ∞

2019-01-01

Mariia Myronova , Marzena Szajewska

MULTIDIMENSIONAL HYBRID BOUNDARY VALUE PROBLEM

2018-12-31

Marzena Szajewska , Agnieszka Tereszkiewicz

The purpose of this paper is to discuss three types of boundary conditions for few families of special functions orthogonal on the fundamental region. Boundary value problems are considered on … The purpose of this paper is to discuss three types of boundary conditions for few families of special functions orthogonal on the fundamental region. Boundary value problems are considered on a simplex F in the real Euclidean space Rn of dimension n &gt; 2.

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Reduction of orbits of finite Coxeter groups of non-crystallographic type

2018-10-01

Zofia Grabowiecka , J. Patera , Marzena Szajewska

A reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices, which transform points of the orbit of any group into points … A reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices, which transform points of the orbit of any group into points of the orbits of its subgroup. Projection matrices and branching rules for orbits of finite Coxeter groups of non-crystallographic type are presented. The novelty in this paper is producing the branching rules that involve non-crystallographic Coxeter groups. Moreover, these branching rules are relevant to any application of non-crystallographic Coxeter groups including molecular crystallography and encryption.

Polytope Contractions within Weyl Group Symmetries

2016-07-16

Marzena Szajewska

A general scheme for constructing polytopes is implemented here specifically for the classes of the most important 3D polytopes, namely those whose vertices are labeled by integers relative to a … A general scheme for constructing polytopes is implemented here specifically for the classes of the most important 3D polytopes, namely those whose vertices are labeled by integers relative to a particular basis, here called the ω-basis. The actual number of non-isomorphic polytopes of the same group has no limit. To put practical bounds on the number of polytopes to consider for each group we limit our consideration to polytopes with dominant point (vertex) that contains only nonnegative integers in ω-basis. A natural place to start the consideration of polytopes from is the generic dominant weight which were all three coordinates are the lowest positive integer numbers. Contraction is a continuous change of one or several coordinates to zero.

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Decomposition matrices for the special case of data on the triangular lattice of SU(3)

2016-07-09

Mark Bodner , J. Patera , Marzena Szajewska

TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

2016-06-30

Marzena Szajewska , Agnieszka Tereszkiewicz

Boundary value problems are considered on a simplex F in the real Euclidean space R2. The recent discovery of new families of special functions, orthogonal on F, makes it possible … Boundary value problems are considered on a simplex F in the real Euclidean space R2. The recent discovery of new families of special functions, orthogonal on F, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on F, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of F a Dirichlet condition is fulfilled and on the other Neumann’s works.

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Faces of root polytopes in all dimensions

2016-05-13

Marzena Szajewska

In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ … In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ n - 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter-Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566-584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.

Group theoretical methods to construct the graphene

2016-01-01

Zofia Grabowiecka , J. Patera , Marzena Szajewska

In this paper, the tiling of the Euclidean plane with regular hexagons whose vertices are occupied by carbon atoms is called the graphene. We describe six different ways to generate … In this paper, the tiling of the Euclidean plane with regular hexagons whose vertices are occupied by carbon atoms is called the graphene. We describe six different ways to generate the graphene by the means of group theory. There are two ways starting from the triangular lattice of Lie algebra $A_2$ and $G_2$, and one way for each of the Lie algebras $B_3$, $C_3$ and $A_3$, by projecting the weight system of their lowest representation to the hexagons of $A_2$. Colouring of the graphene is presented. Changing from one colouring to another is called phase transition. Multistep refinements of the graphene are described.

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Dominant Weight Multiplicities in Hybrid Characters of B n , C n , F 4, G 2

2014-12-22

Francis W. Lemire , J. Patera , Marzena Szajewska

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property … The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions (‘hybrid characters’) is the fact that W(G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the hybrid characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the hybrid characters are determined. The formulas for ‘hybrid dimensions’ are also found for all cases as the zero degree term in power expansion of the ‘hybrid characters’.

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Faces of Platonic solids in all dimensions

2014-06-10

Marzena Szajewska

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 … This paper considers Platonic solids/polytopes in the real Euclidean space 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 types A(n), B(n), C(n), F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groups H3, 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.

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Gaussian curvature of the Bergman metric with weighted Bergman kernel on the unit disc

2012-09-01

Marzena Szajewska

Abstract In the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied. Abstract In the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied.

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Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

2012-05-04

Francis W. Lemire , J. Patera , Marzena Szajewska

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property … The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions (hybrid characters) is the fact that W(G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the characters are determined. The formulas for hybrid dimensions are also found for all cases as the zero degree term in power expansion of the hybrid characters.

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Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

2012-01-01

Francis W. Lemire , J. Patera , Marzena Szajewska

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property … The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions ("hybrid characters") is the fact that W(G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the hybrid characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the hybrid characters are determined. The formulas for "hybrid dimensions" are also found for all cases as the zero degree term in power expansion of the "hybrid characters".

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Four types of special functions ofG2and their discretization

2011-09-27

Marzena Szajewska

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called … Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.

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Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

2010-11-19

Maryna Nesterenko , J. Patera , Marzena Szajewska +1 more

Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases … Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank not greater then 3 are explicitly studied. We derive the polynomials of simple Lie groups B_3 and C_3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

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A property of the curvature and torsion of a regular family of curves in E n

2008-06-01

Marzena Szajewska

Gaussian torsion of a 2-dimensional surface defined implicitly in 4-dimensional Euclidean space

2004-12-31

Yu. A. Aminov , Marzena Szajewska

An expression is found for the Gaussian torsion of a surface F{sup 2} in 4-dimensional Euclidean space E{sup 2} defined implicitly by a system of two equations. An application of … An expression is found for the Gaussian torsion of a surface F{sup 2} in 4-dimensional Euclidean space E{sup 2} defined implicitly by a system of two equations. An application of this formula for the intersection of two quadrics is considered as an example.

Common Coauthors

Coauthor	Papers Together
J. Patera	9
Alina Dobrogowska	4
Francis W. Lemire	3
Agnieszka Tereszkiewicz	3
Zofia Grabowiecka	2
Mariia Myronova	2
Grzegorz Jakimowicz	2
Jiří Hrivnák	1
Mark Bodner	1
Yu. A. Aminov	1
Maryna Nesterenko	1
Karolina Wojciechowicz	1

Commonly Cited References

On discretization of tori of compact simple Lie groups

2009-09-07

Jiří Hrivnák , J. Patera

Three types of numerical data are provided for simple Lie groups of any type and rank. These data are indispensable for Fourier-like expansions of multidimensional digital data into finite series … Three types of numerical data are provided for simple Lie groups of any type and rank. These data are indispensable for Fourier-like expansions of multidimensional digital data into finite series of C- or S-functions on the fundamental domain F of the underlying Lie group G. Firstly, we determine the number |FM| of points in F from the lattice P∨M, which is the refinement of the dual weight lattice P∨ of G by a positive integer M. Secondly, we find the lowest set ΛM of dominant weights, specifying the maximal set of C- and S-functions that are pairwise orthogonal on the point set FM. Finally, we describe an efficient algorithm for finding, on the maximal torus of G, the number of conjugate points to every point of FM. Discrete C- and S-transforms, together with their continuous interpolations, are presented in full generality.

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E-Orbit Functions

2008-01-05

Anatoliy Klimyk

We review and further develop the theory of E-orbit functions.They are functions on the Euclidean space E n obtained from the multivariate exponential function by symmetrization by means of an … We review and further develop the theory of E-orbit functions.They are functions on the Euclidean space E n obtained from the multivariate exponential function by symmetrization by means of an even part W e of a Weyl group W , corresponding to a Coxeter-Dynkin diagram.Properties of such functions are described.They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group W .The E-orbit functions, determined by integral parameters, are invariant with respect to even part W aff e of the affine Weyl group corresponding to W .The E-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform.They also determine a transform on a finite set of points of the fundamental domain F e of the group W aff e (the discrete E-orbit function transform).

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Computation of character decompositions of class functions on compact semisimple Lie groups

1987-05-01

Robert V. Moody , J. Patera

A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group

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Four types of special functions ofG2and their discretization

2011-09-27

Marzena Szajewska

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Gaussian Cubature Arising from Hybrid Characters of Simple Lie Groups

2014-08-19

Robert V. Moody , Lenka Motlochová , J. Patera

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Decomposition of tensor products of the fundamental representations of 𝐸₈

1997-10-17

S. Grimm , J. Patera

Fast recursion formula for weight multiplicities

1982-01-01

Robert V. Moody , J. Patera

The purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple Lie algebras over C. Until … The purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple Lie algebras over C. Until now information about weight multiplicities for all but some special cases [1,2] has had to be found from the recursion formulas of Freudenthal [3] or Racah [4].Typically these formulas become too laborious to use for hand computations for ranks ^ 5 and dimensions ^100 and for ranks -10 and dimensions ~ 10 4 on a large computer [5, 6].With the proposed method the multiplicities can routinely be calculated, even by hand, for dimensions far exceeding these.As an example we present a summary of calculations [7] of all multiplicities in the first sixteen irreducible representations of E s .Let ($ be a semisimple Lie algebra over C with root system A and Weyl group W relative to a Cartan subalgebra §.Let A + be the positive roots with respect to some ordering and II = {a t , ..., a ; } the set of simple roots.Let Q and P be the root and weight lattices respectively spanning the real vector space FC §*.If X C P we denote by X + + the set of dominant elements of X relative to n.Let M be an irreducible ($ -module with highest weight A and weight system £2.An important feature of the approach is the direct determination of £2 ++ without computing outside the dominant chamber.Since every W-orbit is represented by one weight X £ £2 ++ of the same multiplicity, it suffices to compute such X's.The recursion formula for computing the multiplicities is a modification (Proposition 4) of the Freudenthal formula in which the Weyl group has been exploited to collapse it as much as possible.After describing the procedure, we present the E % example.Finally the necessary proofs are given.

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Quasidiagonality of $$C^*$$ C ∗ -Algebras of Solvable Lie Groups

2018-02-01

Ingrid Beltiţă , Daniel Beltiţă

Orthogonal Polynomials Associated with Root Systems

1990-01-01

I. G. Macdonald

Introduction to Lie Algebras and Representation Theory

1972-01-01

James E. Humphreys

Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

2010-11-19

Maryna Nesterenko , J. Patera , Marzena Szajewska +1 more

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Four families of orthogonal polynomials of C2 and symmetric and antisymmetric generalizations of sine and cosine functions

2011-01-01

Lenka Motlochová , J. Patera

Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for … Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction of the polynomials are presented. Orthogonality relations of the four families of polynomials are found together with the appropriate weight fuctions. Tables of the lowest degree polynomials are shown. Numerous trigonometric-like identities are found. Two of the four families of functions are identified as the functions encountered in the Weyl character formula for the finite dimensional irreducible representations of the compact Lie group Sp(4). The other two families of functions seem to play no role in Lie theory so far in spite of their analogous `good' properties.

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Branching rules for the Weyl group orbits of the Lie algebraAn

2009-11-11

M Larouche , Maryna Nesterenko , J. Patera

The orbits of Weyl groups W(An) of simple An-type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming … The orbits of Weyl groups W(An) of simple An-type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W(An) into points of subalgebra orbits are listed for all cases n ⩽ 8 and for the infinite series of algebra–subalgebra pairs An ⊃ An−k−1 × Ak × U1, A2n ⊃ Bn, A2n−1 ⊃ Cn, A2n−1 ⊃ Dn. Numerous special cases and examples are shown.

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Antisymmetric Orbit Functions

2007-02-12

Anatoliy Klimyk

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a … In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

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On the geometry of Lie algebras and Poisson tensors

1994-11-21

José F. Cariñena , Alberto Ibort , G. Marmo +1 more

A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced … A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras which include, as a subclass, semi-simple Lie algebras. A decomposition theorem for Poisson tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower-dimensional real Lie algebras are analysed.

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THE COMPUTATION OF BRANCHING RULES FOR REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS

1977-01-01

Wendy McKay , J. Patera , David Sankoff

On discretization of tori of compact simple Lie groups: II

2012-05-29

Jiří Hrivnák , J. Patera

The discrete orthogonality of special function families, called $C$- and $S$-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivn\'ak and Patera (2009 J. … The discrete orthogonality of special function families, called $C$- and $S$-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivn\'ak and Patera (2009 J. Phys. A: Math. Theor. 42 385208). Here, the results of Hrivn\'ak and Patera are extended to two additional recently discovered families of special functions, called $S^s-$ and $S^l-$functions. The main result is an explicit description of their pairwise discrete orthogonality within each family, when the functions are sampled on finite fragments $F^s_M$ and $F^l_M$ of a lattice in any dimension $n\geq2$ and of any density controlled by $M$, and of the symmetry of the weight lattice of any compact simple Lie group with two different lengths of roots.

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Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

2006-11-01

Robert V. Moody , J. Patera

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Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III

1974-01-01

Tom H. Koornwinder

Semisimple subalgebras of semisimple Lie algebras

1957-01-01

E. B. Dynkin

Three-dimensionalC-,S- andE-transforms

2008-10-20

Maryna Nesterenko , J. Patera

Three-dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three … Three-dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions (C-, S- and E-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region F of the three-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid FM ⊂ F (discrete orthogonality). The positive integer M sets up the density of the lattice containing FM. The expansion of functions given either on F or on FM is the paper's main focus.

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Orthogonal Polynomials of Compact Simple Lie Groups

2011-01-01

Maryna Nesterenko , J. Patera , Agnieszka Tereszkiewicz

Recursive algebraic construction of two infinite families of polynomials in n variables is proposed as a uniform method applicable to every semisimple Lie group of rank n . Its result … Recursive algebraic construction of two infinite families of polynomials in n variables is proposed as a uniform method applicable to every semisimple Lie group of rank n . Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type A 1 . The obtained not Laurent‐type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of types A 1 , A 2 , A 3 , C 2 , C 3 , G 2 , and B 3 together with lowest polynomials.

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Generalization of the concept of classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3442" altimg="si7.svg"><mml:mi>r</mml:mi></mml:math>-matrix to Lie algebroids

2021-03-26

Alina Dobrogowska , Grzegorz Jakimowicz

We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these … We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these fields to construct a collection of interesting new algebroid structures. Next, we show that these constructions can be used in a more general situation, starting from an arbitrary Lie algebroid over M. In the final step, we show that after limiting ourselves to Lie algebras these formulas as a special case contain brackets well known in theory of classical r-matrices. We can think of our constructions as extending the concept of classical r-matrices to Lie algebroids. Several examples illustrate the importance of these constructions.

Characters of Elements of Finite Order in Lie Groups

1984-09-01

Robert V. Moody , J. Patera

In this paper we use the theory of elements of finite order (EFO) as a new and very effective tool for discrete methods in simple Lie groups. The EFO provide … In this paper we use the theory of elements of finite order (EFO) as a new and very effective tool for discrete methods in simple Lie groups. The EFO provide a systematic way of discretely approximating the group. Their character values allow us to systematically determine information about Lie groups and their representations for groups well beyond the range of standard methods. We discuss the theory of EFO, the use of algebraic number fields to single out finite classes of them, and methods of explicitly determining such classes. We introduce an algorithm for effectively computing their character values which utilizes double coset decompositions in the Weyl group and a fast algorithm for determining weight space multiplicities which we developed earlier. The methods are uniform for all simple Lie groups. We briefly discuss a number of applications of this work and finish with a number of tables (including some for $E_6 $) of EFO and their character values.

Discrete Fourier Analysis on Fundamental Domain and Simplex of A d Lattice in d-Variables

2009-10-14

Huiyuan Li , Yuan Xu

Computation of Character Decompositions of Class Functions on Compact Semisimple Lie Groups

1987-04-01

R. V. Moody , J. Patera

A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group K into sums of irreducible characters. The method is based on the use of … A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group K into sums of irreducible characters. The method is based on the use of elements of finite order (EFO) in K and is applicable to a number of problems, including decompositions of tensor products and various symmetry classes of tensors, as well as branching rules in group-subgroup reductions. The main feature is the construction of a decomposition matrix D, computed once and for all for a given range of problems and for a given K, which then reduces any particular splitting to a simple matrix multiplication. Determination of D requires selection of a suitable set S of conjugacy classes of EFO representing a finite subgroup of a maximal torus T of K and the evaluation of (Weyl group) orbit sums on S. In fact, the evaluation of D can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter. The use of the method is illustrated by some extensive examples of tensor product decompositions in ${E_6}$. Modular arithmetic allows all computations to be performed exactly.

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Discrete and continuous cosine transform generalized to Lie groups SU(3) and G(2)

2005-11-01

J. Patera , A. Zaratsyan

In the paper we complete the development and description of the four variants of two-dimensional generalization of the cosine transform started in [Patera and Zaratsyan, J. Math Phys. 46, 053514 … In the paper we complete the development and description of the four variants of two-dimensional generalization of the cosine transform started in [Patera and Zaratsyan, J. Math Phys. 46, 053514 (2005)]. Each variant is based on a compact semisimple Lie group G of rank 2. Here, the groups are SU(3) and G(2). The cosines are generalized as the corresponding C-functions of the Lie group. A C-function is the contribution to an irreducible character from one orbit of the appropriate Weyl group. An explicit description is provided for expansions of functions given on the fundamental region F of the two compact simple Lie groups into series of C-functions. The fundamental region F is an equilateral triangle for SU(3) and half of such a triangle for G(2). Expansion coefficients are calculated using orthogonality of C-functions on F. Discrete expansions are set up on a grid FM⊂F. The grid is defined group theoretically for all positive integers M. It consists of points in F that represent conjugacy classes of elements of the finite maximal Abelian subgroup of G generated by its elements of order M. The C-functions are orthogonal on such a grid; hence, coefficients of discrete expansions are calculated independently of the continuous expansions. Processing digital data, sampled on triangular lattices, is the motivating application here.

A new look at Lie algebras

2023-08-09

Alina Dobrogowska , Grzegorz Jakimowicz

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. … We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Branching rules for Weyl group orbits of simple Lie algebrasBn,CnandDn

2011-02-22

M Larouche , J. Patera

The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of … The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of the maximal reductive subalgebras of B(n), C(n) and D(n). Matrices transforming points of W(B(n)), W(C(n)) and W(D(n)) orbits into points of subalgebra orbits are listed for all cases n<=8 and for the infinite series of algebra-subalgebra pairs B(n) - B(n-1) x U(1), B(n) - D(n), B(n) - B(n-k) x D(k), B(n) - A(1), C(n) - C(n-k) x C(k), C(n) - A(n-1) x U(1), D(n) - A(n-1) x U(1), D(n) - D(n-1) x U(1), D(n) -B(n-1), D(n) - B(n-k-1) x B(k), D(n) -D(n-k) x D(k). Numerous special cases and examples are shown.

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Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

2006-11-08

Robert V. Moody , J. Patera

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry.Three families of class functions are defined on the maximal torus of each compact simply … The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry.Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G.Such functions can always be restricted without loss of information to a fundamental region F of the affine Weyl group.The members of each family satisfy basic orthogonality relations when integrated over F (continuous orthogonality).It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in F (discrete orthogonality), arising as the set of points in F representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus T. The characters of the centre Z of the Lie group allow one to split functions f on F into a sum f = f 1 + • • • + f c , where c is the order of Z, and where the component functions f k decompose into the series of C-, or S-, or E-functions from one congruence class only.

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Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices

1995-06-01

Robert V. Moody , J. Patera

Abstract We give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a … Abstract We give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.

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(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

2007-09-01

A. U. Klimyk , J. Patera

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of … Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms involving the symmetric and antisymmetric multivariate sine and cosine functions.

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Tangent lifts of Poisson and related structures

1995-12-07

Janusz Grabowski , Paweł Urbański

The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector … The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

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Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

2010-12-17

Robert V. Moody , J. Patera

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Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

1992-10-07

Robert V. Moody , J. Patera

The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each … The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each type of face. The representative of a face type is specified by a decoration of the corresponding Coxeter-Dynkin diagram. The rules of domain description are uniform for root lattices of simple Lie groups of all types. An explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple Lie groups. The Coxeter-Dynkin diagrams required here are the diagrams extended by the highest short root. Each diagram is partitioned into two subdiagrams, one describing completely a d-face of the Voronoi domain, its complement completely describing the dual of the d-face. The applicability of the authors' classification method to generalized kaleidoscopes is explained.

On a new relation between semisimple Lie algebras

1980-09-01

J. Patera , R. T. Sharp , R. Slansky

A recently discovered relation between pairs of semisimple Lie algebras is further investigated. This relation, which is called subjoining and denoted by ’’≳’’, is a generalization of inclusion, where a … A recently discovered relation between pairs of semisimple Lie algebras is further investigated. This relation, which is called subjoining and denoted by ’’≳’’, is a generalization of inclusion, where a subalgebra is embedded in an algebra. Nontrivial subjoinings of two algebras of the same type are described. New chains of algebras involving proper inclusions and subjoinings can be formed. Infinite families of maximal subjoinings Cn≳Bn and Bn≳Cn are shown.

Dominant Weight Multiplicities in Hybrid Characters of B n , C n , F 4, G 2

2014-12-22

Francis W. Lemire , J. Patera , Marzena Szajewska

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Wavefronts and reflection groups

1988-06-30

O. P. Shcherbak

CONTENTS Introduction § 1. The manifold of non-regular orbits of the group § 2. and (reflection groups) § 3. and (singularity theory) § 4. The problem of avoiding an obstacle … CONTENTS Introduction § 1. The manifold of non-regular orbits of the group § 2. and (reflection groups) § 3. and (singularity theory) § 4. The problem of avoiding an obstacle § 5. Projections of fronts and normal subgroups of the reflection groups , , § 6. A summary of results on the discriminants of reflection groups References

Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory.

1992-04-01

Eli Passow , T. J. Rivlin

Definitions and Some Elementary Properties Extremal Properties Expansion of Functions in Series of Chebyshev Polynomials Iterative Properties and Some Remarks About the Graphs of the Tn Some Algebraic and Number … Definitions and Some Elementary Properties Extremal Properties Expansion of Functions in Series of Chebyshev Polynomials Iterative Properties and Some Remarks About the Graphs of the Tn Some Algebraic and Number Theoretic Properties of the Chebyshev Polynomials References Glossary of Symbols Index.

Generating functions for G2 characters and subgroup branching rules

1981-12-01

R. Gaskell , R. T. Sharp

The G2 character generator is given; with its help generating functions are derived for branching rules for G2 irreducible representations reduced according to its maximal semisimple subgroups. The G2 character generator is given; with its help generating functions are derived for branching rules for G2 irreducible representations reduced according to its maximal semisimple subgroups.

Character generators for elements of finite order in simple Lie groups A1, A2, A3, B2, and G2

1983-10-01

Robert V. Moody , J. Patera , R. T. Sharp

Elements of finite order in the compact simple Lie groups SU(2), SU(3), SU(4), Sp(4) or O(5), and G(2) are considered. We provide the characters of the elements on irreducible representations … Elements of finite order in the compact simple Lie groups SU(2), SU(3), SU(4), Sp(4) or O(5), and G(2) are considered. We provide the characters of the elements on irreducible representations of Lie groups by assigning appro- priate numerical values to the variables on which the characters of representations of the Lie group depend. In this way we specialize generating functions for the characters of the representations of the Lie groups to the generating functions for characters of the elements of finite order. Particular attention is paid to rational elements, all of whose characters are integers; they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read. Gaussian elements are also studied in detail. Their characters are complex valued with integer real and imaginary parts.

Discrete Fourier Analysis and Chebyshev Polynomials with G2Group

2012-10-03

Huiyuan Li

The discrete Fourier analysis on the 30 0 -60 0 -90 0 triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group … The discrete Fourier analysis on the 30 0 -60 0 -90 0 triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G 2 , which leads to the definition of four families generalized Chebyshev polynomials.The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials.Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials.In particular, their common zeros generate cubature rules of Gauss type.

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On the root system of a coxeter group

1982-01-01

Vinay V. Deodhar

Tangent lifts of bi-Hamiltonian structures

2017-08-01

Alina Dobrogowska , Grzegorz Jakimowicz

We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold … We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (TM,πTM,λ) from a bi-Hamiltonian manifold (M,π1,π2). This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πTM, πTM,λ, π̃Tg*, and π̃Tg*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.

Higher indices of group representations

1976-11-01

J. Patera , R. T. Sharp , P. Winternitz

The nth order index of an irreducible representation of a semisimple compact Lie group, n a nonnegative even integer, is defined as the sum of nth powers of the magnitudes … The nth order index of an irreducible representation of a semisimple compact Lie group, n a nonnegative even integer, is defined as the sum of nth powers of the magnitudes of the weights of the representation. It is shown, in many situations, to have additivity properties similar to those of the dimension under reduction with respect to a subgroup and under reduction of a direct product. The second order index is shown to be Dynkin’s index, multiplied by the rank of the group. Explicit formulas are derived for the fourth order index. A few reduction problems are solved with the help of higher indices as an illustration of their utility.

Faces of Platonic solids in all dimensions

2014-06-10

Marzena Szajewska

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Discrete and continuous graded contractions of Lie algebras and superalgebras

1991-02-07

M. de Montigny , J. Patera

Grading preserving contractions of Lie algebras and superalgebras of any type over the complex number field are defined and studied. Such contractions fall naturally into two classes: the Wigner-Inonu-like continuous … Grading preserving contractions of Lie algebras and superalgebras of any type over the complex number field are defined and studied. Such contractions fall naturally into two classes: the Wigner-Inonu-like continuous contractions and new discrete contractions. A general method is described for any Abelian grading semigroup and any Lie algebra or superalgebra admitting such a grading. All contractions preserving z2-, z3-, and z2*z2-gradings are found. Examples of these gradings and contractions for the simple Lie algebra A2, affine Kac-Moody algebra A(1) and the simple superalgebra osp(2,1) are shown.

On the dependence of the reproducing kernel on the weight of integration

1990-11-01

Zbigniew Pasternak-Winiarski

Inversion of the Pieri formula for Macdonald polynomials

2005-05-25

Michel Lassalle , Michael J. Schlosser

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Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.