Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Symmetries of spherical harmonics


Author: Roberto De Maria Nunes Mendes
Journal: Trans. Amer. Math. Soc. 204 (1975), 161-178
MSC: Primary 22E30; Secondary 43A90
DOI: https://doi.org/10.1090/S0002-9947-1975-0357687-1
MathSciNet review: 0357687
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a group of linear transformations of $ {R^n}$ and $ {H_k}(G)$ the vector space of spherical harmonics invariant under $ G$. The Pálya function is the formal power series $ {\Sigma _{k \geq 0}}{t^k}\dim {H_k}(G)$. In this paper, after classifying all closed subgroups of $ O(4)$, we compute the Pólya functions for these groups. These functions have recently proved to be of interest in quantum mechanics and elementary particle physics.


References [Enhancements On Off] (What's this?)

  • [1] J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [3] -, Éléments de mathématique. Fase. XXX. Algèbre commutative. Chaps. 5, 6, Actualités Sci. Indust., no. 1308, Hermann, Paris, 1964. MR 33 #2660.
  • [4] Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 0072877, https://doi.org/10.2307/2372597
  • [5] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 14, Springer-Verlag, Berlin-Göttingen-New York, 1965. MR 0174618
  • [6] J. Dieudonné, Éléments d'analyse. Tome III, Cahiers Scientifiques, fase. 33, Gauthier-Villars, Paris, 1970. MR 42 #5266.
  • [7] Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0169108
  • [8] Jay P. Fillmore, Symmetries of surfaces of constant width, J. Differential Geometry 3 (1969), 103–110. MR 0247594
  • [9] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge, 1931.
  • [10] Burnett Meyer, On the symmetries of spherical harmonics, Canadian J. Math. 6 (1954), 135–157. MR 0059406
  • [11] G. Pólya and B. Meyer, Sur les symétries des fonctions sphériques de Laplace, C. R. Acad. Sci. Paris 228 (1950), 28-30. MR 10, 281.
  • [12] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0295244

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E30, 43A90

Retrieve articles in all journals with MSC: 22E30, 43A90


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0357687-1
Keywords: Orthogonal group, generating function, spherical harmonic, closed subgroup, invariant
Article copyright: © Copyright 1975 American Mathematical Society