Gaussian cubature arising from hybrid characters of simple Lie groups. (English) Zbl 1317.65070
The authors extend the ideas on multivariate Chebyshev polynomials, nodes, and cubature formulas presented by H. Li and Y. Xu [J. Fourier Anal. Appl. 16, No. 3, 383–433 (2010; Zbl 1194.42006)] and by R. V. Moody and J. Patera [Adv. Appl. Math. 47, No. 3, 509–535 (2011; Zbl 1228.41025)]. This approach leads to Gaussian cubature formulas for families of multivariate polynomials arising from the characters of irreducible representations of any simple Lie group.
Now the authors present new cubature formulas based on the hybrid characters of a simple Lie group. In the short root case, the authors obtain new Gaussian cubature formulas. In the long root case, they preserve new Radau cubature formulas. The nodes for these cubature formulas arise quite naturally from elements of finite order of the Lie group. Finally, the results are illustrated for the Lie group \(G_2\).
Now the authors present new cubature formulas based on the hybrid characters of a simple Lie group. In the short root case, the authors obtain new Gaussian cubature formulas. In the long root case, they preserve new Radau cubature formulas. The nodes for these cubature formulas arise quite naturally from elements of finite order of the Lie group. Finally, the results are illustrated for the Lie group \(G_2\).
Reviewer: Manfred Tasche (Rostock)
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 22E46 | Semisimple Lie groups and their representations |
| 43A80 | Analysis on other specific Lie groups |
| 41A55 | Approximate quadratures |
| 41A63 | Multidimensional problems |
Keywords:
Gaussian cubature formulas; simple Lie group; hybrid characters; Weyl group; Radau cubature formulas; multivariate polynomials; Chebyshev polynomials; Jacobi polynomialsReferences:
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