Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T01:15:28.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Simplicial Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

Barry Monson
Barry Monson*
Affiliation:
University of New Brunswick, Fredericton, New Brunswick
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.

0. Introduction. Simplicial quadratic forms (cf. Definition 1.4), and various equivalent forms, have occasionally been studied in geometry [8], and in number theory [9], [10], in connection with the extremal properties of integral quadratic forms. Our investigations, which employ simple techniques from graph theory and geometry, partly continue both those of Coxeter [5], who introduced the graphs described in Section 1, and Vinberg [20], [21], who described an algorithm for determining a fundamental region for a discrete group acting on spherical, Euclidean, or hyperbolic space. After a preliminary discussion of reflexible forms and the Caley-Klein model for (n − 1)-space (1.2), we define a simplicial form and its graph. Having enumerated them completely, we turn in Section 2 to their equivalence, which is related to a geometric dissection. The unit group for each simplicial form can then be determined from Theorem 3.7.

I wish to thank Professor H. S. M. Coxeter for many helpful ideas, and Professor G. Maxwell and the referee for suggesting numerous improvements.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 1 , 01 February 1983 , pp. 101 - 116
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Bieberbach, L., Uber die Bewegungsgruppen der Euklidischen Rdume, Math. Ann. 70 (1911), 297–336.Google Scholar
2. Bourbaki, N., Groupes etalgèbres de Lie, chap, iv-vi (Hermann, Paris, 1968).Google Scholar
3. Chein, M., Recherche des graphes des matrices de Coxeter hyperboliques d'ordre ≦ 10, Rev. Français Informat. Recherche Operationelle, No. R-3 (1969), 3–16.Google Scholar
4. Coxeter, H. S. M., Discrete groups generated by reflections, Ann. of Math. 35 (1934), 588–621.Google Scholar
5. Coxeter, H. S. M., Extreme forms, Can. J. Math. 8 (1951), 391–441.Google Scholar
6. Coxeter, H. S. M., Non-Euclidean geometry (University of Toronto Press, Toronto, 1965).Google Scholar
7. Coxeter, H. S. M., Regular polytopes (Dover, New York, 1973).Google Scholar
8. Coxeter, H. S. M. and Whitrow, G. S., World structure and non-Euclidean honeycombs, Proc. Royal Soc. London Ser. A. 201 (1950), 417–437.Google Scholar
9. Davenport, H., On a theorem of Markoff, J. London Math. Soc. 22 (1947), 96–99.Google Scholar
10. Davenport, H., On indefinite ternary quadratic forms, Proc. London Math. Soc. (2), 51 (1950), 145–160.Google Scholar
11. Fricke, R. and Klein, F., Vorlesungen uber die Théorie der automorphen Functionen, Vol. 1 (Teubner, Stuttgart, 1897). (Johnson Reprint, New Y'ork, 1965).Google Scholar
12. Johnson, N. W., The theory of uniform polytopes and honeycombs, Ph.D. Thesis, Toronto (1966).Google Scholar
13. Maxwell, G., The crystallography of Coxeter groups, J. Algebra 35 (1975), 159–177.Google Scholar
14. Maxwell, G., On the crystallography of infinite Coxeter groups, Math. Proc. Cambridge Philos. Soc. 82 (1977), 13–24.Google Scholar
15. Maxwell, G., The space groups of two dimensional Minkowski space, Can. J. Math. 30 (1978), 1103–1120.Google Scholar
16. Mennicke, J., Groups of units of ternary quadratic forms, Proc. Roy. Soc. Edinburgh Sect. A 67 (1968), 309–352.Google Scholar
17. Mennicke, J., Pflasterung des dreidimensionalen hyperbolischen Raumes, Math.-Phys. Semesterber. 28 (1980), 55–68.Google Scholar
18. Monson, B., The densities of certain regular star-polytopes, C. R. Math. Rep. Acad. Sci. Canada 2 (1980), 73–78.Google Scholar
19. Vinberg, E. B., Discrete linear groups generated by reflections, Math. USSR-Izv. 5 (1971), 1083–1119.Google Scholar
20. Vinberg, E. B., On groups of unit elements of certain quadratic forms, Math. USSR-Sb. 16, (1972), 17–35.Google Scholar
21. Vinberg, E. B., Some arithmetical discrete groups in Lobacevskil spaces, Discrete subgroups of Lie groups and applications to moduli, Internat. Colloq., Bombay (1973), 323–348. (Oxford Univ. Press, Bombay, 1975).Google Scholar