Tensor power decomposition.<i>B<sub>n</sub></i>case

Kulish, P. P.; Lyakhovsky, V. D.; Postnova, Olga

doi:10.1088/1742-6596/343/1/012095

Cited by 17 publications

(34 citation statements)

References 3 publications

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“…Since the Weyl vector in the space of the fundamental weights is given by the formula ρ = d i=1 λ i , the multiplier Φ as ρ at the denominator of (14), is calculated as…”

Section: Description Of the Methodsmentioning

confidence: 99%

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Damaskinsky¹,

Sokolov²

2017

Theor Math Phys

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“…Since the Weyl vector in the space of the fundamental weights is given by the formula ρ = d i=1 λ i , the multiplier Φ as ρ at the denominator of (14), is calculated as…”

Section: Description Of the Methodsmentioning

confidence: 99%

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Damaskinsky¹,

Sokolov²

2017

Theor Math Phys

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“…The problem of determining M ⊗ p ω ξ is considered below. It was shown in [1], [2] that the functions M ⊗ p ω ξ with different p are connected by the system of recurrence relations…”

Section: Graph Of a Singular Element Of A Module And A Generalizedmentioning

confidence: 99%

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Lyakhovsky

Postnova

2015

Theor Math Phys

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We consider the problem of determining the multiplicity function m ⊗ p ω ξ in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((L ω g ) ⊗p ) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function M ⊗ p ω ξ . This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g, ω) pyramid, i.e., a set of numbers (p, {mi})g,ω satisfying the same system of recurrence relations. We prove that M ⊗ p ω ξ can be represented as a linear combination of the corresponding (p, {mi})g,ω. We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5).

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“…Such polynomials are applied in various areas of mathematics, as well as in physics. Examples of such applications with references can be found in the papers [7][8][9][10][11][12][13][14][15][16].…”

mentioning

confidence: 99%

“…The modification consists of representation of the functions with alternating signs from the Weyl character formula (9) in the form of a difference of two terms (10). This representation allows us to use the calculation scheme presented in this section.…”

mentioning

confidence: 99%

Generating Functions of Chebyshev Polynomials in Three Variables

Sokolov

2016

J Math Sci

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In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A 3 Lie algebra. Bibliography: 21 titles.To Petr Petrovich Kulish on the occasion of his 70th birthday 1. The aim of the present paper is to obtain generating functions of Chebyshev polynomials in three variables. Chebyshev polynomials in several variables associated with the root systems of simple Lie algebras were intensively studied in the last decades [1][2][3][4][5][6]. Such polynomials are applied in various areas of mathematics, as well as in physics. Examples of such applications with references can be found in the papers [7][8][9][10][11][12][13][14][15][16].Multivariate Chebyshev polynomials are natural generalizations of the classical ones. The classical Chebyshev polynomials T n (x) of the first kind are defined by the following formula (see, for example, [17]):( 1 ) they satisfy the following three-term recurrence relation:The classical Chebyshev polynomials U n (x) of the second kind are defined by the formulawhich is different from (1), but they satisfy the same recurrence relation (2). The initial conditions for the above polynomials are T 0 (x) = 1, T 1 (x) = x, U 0 (x) = 1, and U 1 (x) = 2x;( 3 ) together with the recurrence relation (2), they determine the polynomials uniquely without reference to (1) and (2). It is known that the function cos nφ can be treated as the invariant mean of the exponential function over the Weyl group of the A 1 algebra root system:Transition to a function that is the invariant mean over the Weyl group W of another root system leads us to generalization of the classical Chebyshev polynomials to polynomials in several variables [1][2][3][4]. Let us shortly recall the way in which such W -invariant functions can be constructed. Let R(L) be a reduced root system of any simple Lie algebra L. Such a system is a set of vectors in the d-dimensional Euclidean space E d supplied with scalar product (., .). The system R is uniquely determined by the base of simple roots α i , i = 1, . . . , d, and by a finite group W (R) which is generated by base reflections. This group is called the Weyl group.

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Tensor power decomposition.B_ncase

Cited by 17 publications

References 3 publications

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Generating Functions of Chebyshev Polynomials in Three Variables

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Tensor power decomposition.Bncase

Cited by 17 publications

References 3 publications

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

Generating Functions of Chebyshev Polynomials in Three Variables

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Tensor power decomposition.B_ncase