Geodesic
Reflections and Three-Dimensionality
Matematičeskie zametki, Tome 73 (2003) no. 6, pp. 848-852 
V. A. Kolmykov. Reflections and Three-Dimensionality. Matematičeskie zametki, Tome 73 (2003) no. 6, pp. 848-852. http://geodesic.mathdoc.fr/item/MZM_2003_73_6_a5/
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     title = {Reflections and {Three-Dimensionality}},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_6_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Coxeter H. S. M., “The product of generators of finite group generated by reflections”, Duke Math. J., 1951, no. 18, 765–782 | DOI | MR | Zbl

[2] Burbaki N., Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Gruppy, porozhdennye otrazheniyami. Sistemy kornei, Per. s frants., Mir, M., 1972 | Zbl

[3] Bernshtein I. N., Gelfand I. M., Ponomarev V. A., “Funktory Kokstera i teorema Gabrielya”, UMN, 28:2 (1973), 19–33 | MR

[4] Kolmykov V. A., “Preobrazovanie Kokstera i chislo "$-1$"”, Matem. sb., 191:10 (2000), 51–56 | MR | Zbl