Abstract
Lie groups with two different root lengths allow two ‘mixed sign’ homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.
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Acknowledgments
We gratefully acknowledge the support of this work by the Natural Sciences and Engineering Research Council of Canada and by the MIND Research Institute of Irvine, Calif. L.M. would also like to express her gratitude to the Centre de recherches mathématiques, Université de Montréal, for the hospitality extended to her during her doctoral studies as well as to the Institute de Sciences Mathématiques de Montréal and Foundation J.A. DeSève for partial support of her studies. We are also grateful to the referees of this paper who provided us with much insight into the historical background and literature in the area.
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Communicated by Arieh Iserles.
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Moody, R.V., Motlochová, L. & Patera, J. Gaussian Cubature Arising from Hybrid Characters of Simple Lie Groups. J Fourier Anal Appl 20, 1257–1290 (2014). https://doi.org/10.1007/s00041-014-9355-0
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DOI: https://doi.org/10.1007/s00041-014-9355-0