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Register a new userOrthogonal polynomials of compact simple Lie groups: Branching rules for polynomials
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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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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 proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. 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.
The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of $n$ variables based on the root lattices of compact simple Lie groups $G$ of any type and of rank $n$. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from $A$-type lattices, yield Gaussian cubature formulae for each simple Lie group $G$ based on interpolation points that arise from regular elements of finite order in $G$. The polynomials arise from the irreducible characters of $G$ and the interpolation points as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections.
Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,...,t_r$, where r (=1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R=S is of type $A_n$, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.
We consider the unital associative algebra $mathcal{A}$ with two generators $mathcal{X}$, $mathcal{Z}$ obeying the defining relation $[mathcal{Z},mathcal{X}]=mathcal{Z}^2+Delta$. We construct irreducible tridiagonal representations of $mathcal{A}$. Depending on the value of the parameter $Delta$, these representations are associated to the Jacobi matrices of the para-Krawtchouk, continuous Hahn, Hahn or Jacobi polynomials.
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.
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