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Some applications of representation theory

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Representation Theory of Finite Groups and Finite-Dimensional Algebras

Part of the book series: Progress in Mathematics ((PM,volume 95))

Abstract

Group representation theory has grown out of group theory. Therefore one way of measuring progress is by looking at the applications of representation theory to group theory and theories where groups play a role. In this note I discuss two areas of applications, which I have been involved in. A third area, which is left out here, is that of extensions. As pointed out in [HoP 89] representation theory not only provides abelian normal subgroups for extensions with given factor group, but also nonabelian (e.g. pro-p) normal subgroups. A fourth area, which I also only mention in passing, is that of applications to computational group theory, more precisely to the investigation of finite presentations of groups, cf. [Ple 87], [HoP 89] Chapter 7, [HoP 90].

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References

  1. Ei. Bannai, Et. Bannai, On some finite subgroups of GL(n, Q). J. Fac. Sci. Univ. Tokyo, Sect. IA 20,(3) (1973), 319–340.

    MathSciNet  MATH  Google Scholar 

  2. H. F. Blichfeldt, Finite Collineation Groups. Chicago: University of Chicago Press 1917.

    Google Scholar 

  3. R. Brauer, Über endliche lineare Gruppen von Primzahlgrad. Math. Annalen 169, 73–96 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley 1977.

    Google Scholar 

  5. R. Bülow, Über Dadegruppen in GL(5, Z). Dissertation RWTH Aachen 1973.

    Google Scholar 

  6. W. Burnside, The determination of all groups of rational linear substitutions of finite order which contain the symmetric group in the variables. Proc. London Math. Cos. (2) 10 (1912), 284–308.

    Article  Google Scholar 

  7. P.J. Cameron, Finite Permutation Groups and Finite Simple Groups. Bull. London Math. Soc. 13 (1981), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  8. J.H. Conway, N.J.A. Sloane, Low-dimensional lattices. II. Subgroups of GL(n, Z). Proc. R. Soc. London A 419, 29–68 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Oxford University Press 1985.

    Google Scholar 

  10. C.W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras. Interscience, New York, 1962.

    MATH  Google Scholar 

  11. E.C. Dade, Integral Systems of Imprimitivity. Math. Annalen 154, 383–386 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  12. E.C. Dade, The maximal finite groups of 4 × 4 matrices. Ill. J. Math. 9 (1965), 99–122.

    MathSciNet  MATH  Google Scholar 

  13. W. Feit, On integral representations of finite groups. Proc. London Math Soc. (3) 29 (1974), 633–683.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Feit, On finite linear groups in dimension at most 10. 397–407 in Proceedings Conference on Finite Groups (ed W. R. Scott and F. Gross), New York Academic Press 1976.

    Google Scholar 

  15. D. F. Holt, W. Plesken, Perfect Groups. Oxford University Press 1989.

    Google Scholar 

  16. D. F. Holt, W. Plesken, A cohomological criterion for a finitely presented group to be infinite, submitted 1990.

    Google Scholar 

  17. C. Jordan, Mémoire sur l’équivalence des formes. J. Ecole Polytech. 48 (1880), 112–150. Oeuvres de C. Jordan, Vol.III, Gauthier-Villars, Paris, 1962, 421–460.

    Google Scholar 

  18. M. Kneser, Zur Theorie der Kristallgitter. Math. Ann. 127, 105–106 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Landazuri, G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. Journal of Algebra 32, 418–443 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Plesken, M. Pohst, On maximal finite irreducible subgroups of GL(n, Z). I. The five-and seven-dimensional case. II. The six-dimensional case, Math. Comp. 31 (138)(1977), 536–577; III. The nine-dimensional case. IV. Remarks on even dimensions with applications to n = 8. V. The eight-dimensional case and a complete description of dimensions less than ten, Math. Comp. 34 (149) (1980), 245–301.

    MathSciNet  MATH  Google Scholar 

  21. W. Plesken On reducible and decomposable representations of orders. J. reine angew. Math. 297 (1978), 188–210.

    MathSciNet  MATH  Google Scholar 

  22. W. Plesken, Bravais groups in low dimensions. Proceedings of a Conference on Crystallographic Groups, Bielefeld 1979. match 10 (1981), 97–119.

    Google Scholar 

  23. W. Plesken, Finite unimodular groups of prime degree and circulants. J. of Algebra vol. 97, no. 1 (1985), 286–312.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Plesken, Towards a Soluble Qutient Algorithm. J. Symbolic Computation (1987) 4, 111–122.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Plesken, W. Hanrath, The lattices of six-dimensional euclidean space. Math, of Comput. vol. 43, no. 168 (1984), 573–587.

    Article  MathSciNet  MATH  Google Scholar 

  26. S.S. Ryskov, On maximal finite groups of integer n × n-matrices. Dokl. Akad. Nauk SSSR 204 (1972), 561–564. Sov. Math. Dokl. 13 (1972), 720–724.

    MathSciNet  Google Scholar 

  27. S.S. Ryskov, Maximal finite groups of integral n × n matrices and full groups of integral automorphisms of positive quadratic forms (Bravais models). Tr. Mat. Inst. Steklov 128 (1972), 183–211. Proc. Steklov Inst. Math. 128 (1972), 217–250.

    MathSciNet  Google Scholar 

  28. S.S. Ryskov, Z.D. Lomakina, Proof of a theorem on maximal finite groups of integral 5 × 5-matrices. Proc. Steklov Inst. Math. (1982) Issue 1, 225–235.

    Google Scholar 

  29. W. Scharlau, Quadratic and Hermitian forms. Springer-Verlag 1985.

    Google Scholar 

  30. B. Souvignier, Irreduzible Bravaisgruppen. Diplomarbeit Aachen 1991.

    Google Scholar 

  31. H. Zassenhaus, Neuer Beweis der Endlichkeit der Klassenzahl bei unimodularer Äquivalenz endlicher ganzzahliger Substitutionsgruppen. Hamb. Abh. 12 (1938), 276–288. Additional References for Chapter III

    Article  Google Scholar 

Additional References for Chapter III

  1. L. Auslander, M. Kuranishi, On the holonomy group of locally Euclidean spaces. Ann. Math. 65, 411–415 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  2. L.S. Charlap, Bieberbach groups and flat manifolds. Berlin Heidelberg New Zork: Springer 1986.

    Book  MATH  Google Scholar 

  3. G. Cliff, A. Weiss, Torsion free space groups and permutation lattices for finite groups. Contemporary Mathem. vol 93 (Representation Theory, Group Rings, and Coding Thery) AMS (1989), 123–132.

    Google Scholar 

  4. G. Hiss, A. Szczepanski, On torsion free crystallographic groups. submitted 1990.

    Google Scholar 

  5. H. Hiller, C.-H. Sah, Holonomy of flat manifolds with b 1 = 0. Q.J. Math., Oxf. II Ser. 37, 177–187 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Meyer, Minimal extensions of finite groups by free abelian groups. Arch. Math. 42, 16–31 (1984).

    Article  MATH  Google Scholar 

  7. W. Plesken, Minimal dimensions for flat manifolds with prescribed holonomy. Math. Ann. 284, 477–486 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Szczepanski, Five dimensional Bieberbach groups with trivial centre. submitted 1990.

    Google Scholar 

  9. J.A. Wolf, Spaces of constant curvature. New York: McGraw-Hill 1967.

    MATH  Google Scholar 

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Authors
  1. W. Plesken
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Editors and Affiliations

  1. Institut für Experimentelle Mathematik, Universität GHS Essen, Ellernstrasse 29, D-4300, Essen 12, Germany

    G. O. Michler

  2. Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse, D-4800, Bielefeld, Germany

    C. M. Ringel

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© 1991 Springer Basel AG

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Plesken, W. (1991). Some applications of representation theory. In: Michler, G.O., Ringel, C.M. (eds) Representation Theory of Finite Groups and Finite-Dimensional Algebras. Progress in Mathematics, vol 95. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8658-1_22

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  • DOI: https://doi.org/10.1007/978-3-0348-8658-1_22

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9720-4

  • Online ISBN: 978-3-0348-8658-1

  • eBook Packages: Springer Book Archive

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