Abstract
If the dimension of a linear space is not greater than 3, then the characteristic polynomial of the Coxeter transformation associated with any symmetric matrix is invariant under the natural action of the symmetric group. If the dimensionality is greater than 3, then this statement does not hold. The set of all trees such that the spectrum of their associated Coxeter transformation contains negative one is three-dimensional.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
H. S. M. Coxeter, “The product of generators of finite group generated by reflections,” Duke Math. J. (1951), no. 18, 765–782.
N. Bourbaki, Éléements de mathématique. Fasc. XXXIV, Groupes et algèbres de Lie, Chapitre IV–IV, Hermann, Paris, 1968.
I. N. Bernshtein, I. M. Gel'fand, and V. A. Ponomarev, “Coxeter functors and Gabriel's theorem,” Uspekhi Mat.Nauk [Russian Math. Surveys], 28 (1973), no. 2, 19–33.
V. A. Kolmykov, “Coxeter transformation and the number “–1”,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 191 (2000), no. 10, 51–56.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kolmykov, V.A. Reflections and Three-Dimensionality. Mathematical Notes 73, 802–805 (2003). https://doi.org/10.1023/A:1024097729426
Issue Date:
DOI: https://doi.org/10.1023/A:1024097729426