Theoretica chimica acta

, Volume 91, Issue 5–6, pp 291–314 | Cite as

Dominant representations and a markaracter table for a group of finite order

  • Shinsaku Fujita


The concept of markaracter is proposed to discuss marks and characters for a group of finite order on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominant representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers of fixed points corresponding to each DR are collected to form a row vecter called a dominant markaracter (mark-character). Such dominant markaracters for the SDR are collected as a markaracter table. The markaracter table is related to a subdominant markaracter table of its subgroup so that the corresponding row of the former table is constructed from the latter. The data of the markaracter table are in turn used to construct a character table of the group, after each character is regarded as a markaracter and transformed into a multiplicity vector. The concept of orbit index is proposed to classify multiplicity vectors; thus, the orbit index of each DR is proved to be equal to one, while that corresonding to an irreducible representation is equal to zero.

Key words

Dominant representation Markaracter Group Orbit index 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cotton FA (1971) Chemical Applications of Group Theory. Wiley, New YorkGoogle Scholar
  2. 2.
    Jaffé HH, Orchin M (1965) Symmetry in Chemistry. Wiley, ChichesterGoogle Scholar
  3. 3.
    Hall LH (1969) Group Theory and Symmetry in Chemistry. McGraw-Hill, New YorkGoogle Scholar
  4. 4.
    Bishop DM (1973) Group Theory and Chemistry. Clarendon, OxfordGoogle Scholar
  5. 5.
    Kettle SFA (1985) Symmetry and Structure. Wiley, ChichesterGoogle Scholar
  6. 6.
    Ladd MFC (1989) Symmetry in Molecules and Crystals. Ellis Horwood, ChichesterGoogle Scholar
  7. 7.
    Harris DC, Bertolucci MD (1989) Symmetry and Spectroscopy. Dover, New YorkGoogle Scholar
  8. 8.
    Hargittai I, Hargittai H (1986) Symmetry through the Eyes of a Chemist. VCH, WeinheimGoogle Scholar
  9. 9.
    Burnside W (1911) Theory of Groups of Finite Order. 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  10. 10.
    Sheehan J (1968) Can. J. Math. 20:1068–1076Google Scholar
  11. 11.
    Kerber A, Thürlings KJ (1982) In: Jngnickel D, Vedder K (eds) Combinatorial Theory. Springer, Berlin, pp 191–211Google Scholar
  12. 12.
    Redfield JH (1984) J. Graph Theory 8:205–223Google Scholar
  13. 13.
    Fujita S (1993) J. Math. Chem. 12:173–195Google Scholar
  14. 14.
    Fujita S (1994) J. Graph Theory 18:349–371Google Scholar
  15. 15.
    Hässelbarth W (1985) Theor. Chim. Acta 67:339–367Google Scholar
  16. 16.
    Mead CA (1987) J. Am. Chem. Soc. 109:2130–2137Google Scholar
  17. 17.
    Fujita S (1989) Theor. Chim. Acta 76:247–268Google Scholar
  18. 18.
    Fujita S (1991) Symmetry and Combinatorial Enumeration in Chemistry. Springer, Berlin-HeidelbergGoogle Scholar
  19. 19.
    Fujita S (1992) Theor. Chim. Acta 82:473–498Google Scholar
  20. 20.
    Lloyd EK (1992) J. Math. Chem. 11:207–222Google Scholar
  21. 21.
    Fujita S (1990) Theor. Chim. Acta 78:45–63Google Scholar
  22. 22.
    Fujita S (1990) J. Math. Chem. 5:99–120Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Shinsaku Fujita
    • 1
  1. 1.Ashigara Research LaboratoriesFuji Photo Film Co., Ltd.KanagawaJapan

Personalised recommendations