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Orthogonal, symplectic and unitary representations of finite groups


Author: Carl R. Riehm
Journal: Trans. Amer. Math. Soc. 353 (2001), 4687-4727
MSC (2000): Primary 20C99, 11E08, 11E12; Secondary 20G05, 51F25
Published electronically: June 27, 2001
MathSciNet review: 1852079
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Abstract: Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are:

I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant.

II. If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group).

III. Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically'' equivalent.

This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent.

We assume throughout that the characteristic of $K$ does not divide $2\vert G\vert$.

Solutions to I and II are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is ``split''. The results for III are strongest when the degrees of the absolutely irreducible representations of $G$ are odd - for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that $K$ is a local or global field, when the representations are split.


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  • 1. N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Sci. Ind. no. 1261, Hermann, Paris, 1958 (French). MR 0098114
  • 2. Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • 3. Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979
  • 4. Andreas W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. (2) 102 (1975), no. 2, 291–325. MR 0387392
  • 5. Martin Epkenhans, Spurformen über lokalen Körpern, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 44, Universität Münster, Mathematisches Institut, Münster, 1987 (German). MR 890958
  • 6. Martin Epkenhans, Trace forms of normal extensions over local fields, Linear and Multilinear Algebra 24 (1989), no. 2, 103–116. MR 1007248, 10.1080/03081088908817903
  • 7. Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
  • 8. A. Fröhlich and A. M. McEvett, Forms over rings with involution, J. Algebra 12 (1969), 79–104. MR 0274480
  • 9. A. Fröhlich and A. M. McEvett, The representation of groups by automorphisms of forms, J. Algebra 12 (1969), 114–133. MR 0240217
  • 10. Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
  • 11. Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779
  • 12. H. Koch, Algebraic number theory, Translated from the 1988 Russian edition, Springer-Verlag, Berlin, 1997. Reprint of the 1992 translation. MR 1474965
  • 13. A. I. Malcev, On semi-simple subgroups of Lie groups, Amer. Math. Soc. Translation 1950 (1950), no. 33, 43. MR 0037848
  • 14. A. M. McEvett, Forms over semisimple algebras with involution, J. Algebra 12 (1969), 105–113. MR 0274481
  • 15. John Milnor, On isometries of inner product spaces, Invent. Math. 8 (1969), 83–97. MR 0249519
  • 16. Gabriele Nebe, Invariants of orthogonal G-modules from the character table, Experimental Mathematics 9 (2000), 623-629. CMP 2001:06
  • 17. Gabriele Nebe, Orthogonal Frobenius reciprocity, J. Algebra 225 (2000), no. 1, 250–260. MR 1743660, 10.1006/jabr.1999.8114
  • 18. Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859
  • 19. O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Bd. 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0152507
  • 20. I. Reiner, Maximal orders, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975. London Mathematical Society Monographs, No. 5. MR 0393100
  • 21. Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063
  • 22. Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380
  • 23. C. T. C. Wall, Classification of Hermitian Forms. VI. Group rings, Ann. of Math. (2) 103 (1976), no. 1, 1–80. MR 0432737

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Additional Information

Carl R. Riehm
Affiliation: Deptartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1
Email: riehm@mcmaster.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02807-0
Received by editor(s): April 5, 2000
Received by editor(s) in revised form: October 30, 2000
Published electronically: June 27, 2001
Article copyright: © Copyright 2001 American Mathematical Society