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A Discrete Realization of the Higher Rank Racah Algebra
Constructive Approximation ( IF 2.7 ) Pub Date : 2019-07-11 , DOI: 10.1007/s00365-019-09475-0
Hendrik De Bie , Wouter van de Vijver

In previous work, a higher rank generalization R ( n ) of the Racah algebra was defined abstractly. The special case of rank one encodes the bispectrality of the univariate Racah polynomials and is known to admit an explicit realization in terms of the operators associated with these polynomials. Starting from the Dunkl model for which we have an action by R ( n ) on the Dunkl-harmonics, we show that connection coefficients between bases of Dunkl-harmonics diagonalizing certain Abelian subalgebras are multivariate Racah polynomials. By lifting the action of R ( n ) to the connection coefficients, we identify the action of the Abelian subalgebras with the action of the Racah operators defined by J. S. Geronimo and P. Iliev. Making appropriate changes of basis, one can identify each generator of R ( n ) as a discrete operator acting on the multivariate Racah polynomials.

中文翻译:

高阶 Racah 代数的离散实现

在之前的工作中,抽象地定义了 Racah 代数的更高阶泛化 R ( n )。秩 1 的特殊情况对单变量 Racah 多项式的双谱性进行编码,并且已知根据与这些多项式相关联的运算符允许显式实现。从 Dunkl 模型开始,我们对 Dunkl 谐波有 R ( n ) 的作用,我们证明了 Dunkl 谐波对角化某些阿贝尔子代数的基之间的连接系数是多元 Racah 多项式。通过将 R ( n ) 的作用提升到连接系数,我们用 JS Geronimo 和 P. Iliev 定义的 Racah 算子的作用来识别阿贝尔子代数的作用。适当改变基础,
更新日期:2019-07-11
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