Theoretica chimica acta

, Volume 70, Issue 3, pp 203–219 | Cite as

The method of ascending symmetry for irreducible characters of finite groups

  • Noam Agmon


The irreducible characters of a finite group are determined uniquely by those of a minimal set of maximal subgroups. The method is based on the construction of all class functions which are irreducible characters on every maximal subgroup. These are generalized characters by a theorem of Brauer, so that the irreducible characters are obtained by checking the norm. An alternative characterization of irreducible characters, the Maximum Mixing Rule, works for all point symmetry groups, and its physical significance is discussed. As an example, the character tables for all point symmetry groups and crystal double-groups are constructed in this way.

Key words

Character tables Finite groups Generalized and irreducible characters Symmetry Maximal subgroups Mixing 


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References and notes

  1. 1.
    Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New YorkGoogle Scholar
  2. 2.
    Weyl H (1931) Theory of groups and quantum mechanics. Princeton University Press, Princeton, NJGoogle Scholar
  3. 3.
    Lomont JS (1959) Applications of finite groups. Academic Press, New YorkGoogle Scholar
  4. 4.
    Hamermesh M (1962) Group theory and its applications to physical problems. Addison-Wesley, Reading, MassGoogle Scholar
  5. 5.
    Tinkham M (1964) Group theory and quantum mechanics. McGraw-Hill, New YorkGoogle Scholar
  6. 6.
    Jansen L, Boon M (1967) Theory of finite groups. Applications in physics. North-Holland, AmsterdamGoogle Scholar
  7. 7.
    Miller W Jr (1972) Symmetry groups and their applications. Academic Press, New YorkGoogle Scholar
  8. 8.
    Schensted IV (1976) A course on the application of group theory to quantum mechanics. NEO Press, Peaks Island, MaineGoogle Scholar
  9. 9.
    Wilson EB Jr, Decius JC, Cross PC (1955) Molecular vibrations. McGraw-Hill, New YorkGoogle Scholar
  10. 10.
    Cotton FA (1971) Chemical applications of group theory, 2nd edn. Wiley, New YorkGoogle Scholar
  11. 11. a)
    McWeeney R (1963) Symmetry: an introduction to group theory and its applications. Pergamon Press, London;Google Scholar
  12. 11. b)
    Chisholm CDH (1976) Group theoretical techniques in quantum chemistry. Academic Press, LondonGoogle Scholar
  13. 12.
    Littlewood DE (1950) The theory of group characters and matrix representations of groups, 2nd edn. Oxford University Press, LondonGoogle Scholar
  14. 13.
    Boerner H (1969) Representations of groups, with special consideration for the need of modern physics, 2nd edn. North-Holland, AmsterdamGoogle Scholar
  15. 14.
    Dornhoff L (1971) Group representation theory, part A. Marcel-Dekker, New YorkGoogle Scholar
  16. 15.
    Isaacs IM (1976) Character theory of finite groups. Academic Press, New YorkGoogle Scholar
  17. 16.
    Neubüser J (1982) Physica 114A:493; Neubüser J, Pahlings H, Plesken W (1984) In: Atkinson MD (ed) Computational group theory. Academic Press, London, p 195Google Scholar
  18. 17.
    Rotman JJ (1984) An introduction to the theory of groups, 3rd edn. Allyn and Bacon, BostonGoogle Scholar
  19. 18. a)
    Wigner EP (1971) Proc. Roy. Soc. A322:181Google Scholar
  20. 18. b)
    Karlof J (1975) Trans Am Math Soc 207:329Google Scholar
  21. 18. c)
    Hurley AC (1982) Chem Phys Lett 91:163Google Scholar
  22. 19.
    Aczél J, Daróczy Z (1975) On measures of information and their characterization. Academic Press, New YorkGoogle Scholar
  23. 20.
    Callen HB (1960) Thermodynamics. Wiley, New YorkGoogle Scholar
  24. 21.
    Ruch E (1964) Theor Chim Acta 2:193Google Scholar
  25. 22.
    Biedenharn LC, Brouwer W, Sharp WT (1968) Rice University Studies 54:2Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Noam Agmon
    • 1
  1. 1.Department of Physical Chemistry and the Fritz Haber Center for Molecular DynamicsThe Hebrew UniversityJerusalemIsrael

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