Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T00:47:58.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial Invariants of Finite Linear Groups of Degree Two

Published online by Cambridge University Press:  20 November 2018

W. Cary Huffman
W. Cary Huffman*
Affiliation:
Loyola University, Chicago, Illinois
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recently invariant theory of linear groups has been used to determine the structure of several weight enumerators of codes. Under certain conditions on the code, the weight enumerator is invariant under a finite group of matrices. Once all the polynomial invariants of this group are known, the form of the weight enumerator is restricted and often useful results about the existence and structure of codes can be found. (See [5], [8], [14], and [15].) Many of the groups in these applications are of degree 2; in this paper all the invariants of finite 2 X 2 matrix groups over C are determined.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 317 - 330
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Blichfeldt, H. F., Finite collineation groups (University of Chicago Press, Chicago, 1917).Google Scholar
2. Burnside, W., Theory of groups of finite order (Dover, New York, 1955).Google Scholar
3. Coxeter, H. S. M., Regular complex poly topes (Cambridge University Press, 1974).Google Scholar
4. DuVal, P., Homographies, quaternions and rotations (Oxford: Oxford University Press, 1964).Google Scholar
5. Gleason, A. M., Weight polynomials of self-dual codes and the MacWilliams identities, in: Actes Congrès Internat. Math. 1970, Vol. 3, Gauthiers-Villars, Paris (1971), 211–215.Google Scholar
6. Huffman, W. C. and Sloane, N. J. A., Most primitive groups have messy invariants, to appear in Adv. in Math.Google Scholar
7. Klein, F., Lectures on the icosahedron and the solution of equations of the fifth degree (Dover, New York, 1956).Google Scholar
8. 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.Google Scholar
9. Mallows, C. L. and Sloane, N. J. A., On the invariants of a linear group of order 336, Proc. Camb. Phil. Soc. 74 (1973), 435–440.Google Scholar
10. Miller, G. A., Blichfeldt, H. F., and Dickson, L. E., Theory and applications of finite groups (Dover, New York, 1961).Google Scholar
11. Molien, T., Uber die Invarianten der lineare Substitutionsgruppe, Sitzungsber. Kônig. Preuss. Akad. Wiss. (1897), 1152–1156.Google Scholar
12. Riemenschneider, O., Die Invarianten der endlichen Untergruppen von GL(2, C), Math Zeit. 153 (1977), 37–50.Google Scholar
13. Shephard, G. C., and Todd, J. A., Finite unitary reflection groups, Can. J. Math. 6 (1954), 274–304.Google Scholar
14. Sloane, N. J. A., Error-correcting codes and invariant theory: new applications of a nineteenthcentury technique, Amer. Math. Monthly 84 (1977), 82–107.Google Scholar
15. Sloane, N. J. A., Weight enumerators of codes, in Combinatorics (Proceedings of the NATO Advanced Study Institute, July, 1974), 115–142.Google Scholar
16. Stanley, R., Relative invariants of finite groups generated by pseudoreflections, J. Algebra 49 (1977), 134–148.Google Scholar