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On the invariants of a linear group of order 336

Published online by Cambridge University Press:  24 October 2008

C. L. Mallows  and
N. J. A. Sloane
C. L. Mallows
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
N. J. A. Sloane
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
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Abstract

The polynomial invariants of a certain classical linear group of order 336 arise naturally in studying error-correcting codes over GF(7). An incomplete description of these invariants was given by Maschke in 1893. With the aid of the Poincaré series for this group, found by Edge in 1947, we complete Maschke's work by giving a unique representation for the invariants in terms of 12 basic invariants. A conjecture is made concerning the relationship between the Poincaré series and the degrees of the basic invariants for any linear group. A partial answer to this conjecture, due to E. C. Dade, is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 3 , November 1973 , pp. 435 - 440
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Atiyah, M. F. and MacDonald, I. G.Introduction to commutative algebra (Reading, Massachusetts, Addison-Wesley, 1969), p. 116.Google Scholar
(2)Baker, H. F.Note introductory of the study of Klein's group of order 168. Proc. Cambridge Philo8. Soc. 31, (1935), 468–481.CrossRefGoogle Scholar
(3)Brioshci, F.Über die Jacobi'sche Modulargleichung vom achtenGrade. Math. Ann. 15 (1879), 241–250.CrossRefGoogle Scholar
(4)Brown, W. S.Altran user's manual (Bell Laboratories, Murray Hill, New Jersey, 1971).Google Scholar
(5)Burnside, W.Theory of groups of finite order, 2nd ed. (1911), republished by Dover Publications, New York, 1955.Google Scholar
(6)Dade, E. C. Written communication, 23 04 1972.Google Scholar
(7)Edge, W. L.The Klein group in three dimensions. Acta Math. 79 (1947), 153–223.CrossRefGoogle Scholar
(8)Gleason, A. M. Weight polynomials of self-dual codes and the MacWilliams identities. Actes, Congrèe intern. Math., 1970 (Gauthier-Villars, 1971), tome 3, pp. 211–215.Google Scholar
(9)Hall, A. D. JrThe Altran system for rational function manipulation – A survey. Comm. ACM 14 (1971), 517–521.CrossRefGoogle Scholar
(10)Klein, F.Über die Auflösung gewisser Gleichungen vom seibenten und achten Grade. Math. Ann. 15 (1879), 25–82.CrossRefGoogle Scholar
(11)MacWilliams, F. J., Mallows, C. L. and Sloane, N. J. A.Generalizations of Gleason's theorem on weight enumerators of self-dual codes. IEEE Trans. information Theory IT-18 (1972), 794–805.CrossRefGoogle Scholar
(12)Maschke, H. The invariants of a group of 2.168 linear quaternary substitutions. Inter national Mathematical Congress 1893 (New York, Macmillan, 1896), pp. 175–186.Google Scholar
(13)MoLien, T.Über die Invarianten der linearen Substitutions-gruppen. Sitz. König. Preuss. Akad. Wiss. (1897), 1152–1156.Google Scholar
(14)Shephard, G. C. and Todd, J. A.Finite unitary reflection groups. Canad. J. Math. 6 (1954), 274–304.CrossRefGoogle Scholar