Abstract
The goal of this chapter is twofold. On one hand we analyze integral quadratic forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) such that there is a basis in \(\mathbb {Z}^n\) in which q is unitary (that is, all diagonal coefficients are equal to one). Such quadratic forms are called concealed. Some methods to identify concealed forms are discussed, for instance, in the positive case we make use of spectral properties of graphs. On the other hand we study certain subgroups of the group of isometries associated to unitary forms, so called Weyl groups. Spectral properties of Coxeter transformations are presented, as well as some relations of cyclotomic polynomials with Dynkin and extended Dynkin diagrams. Further properties of Coxeter matrices are considered, including periodicity phenomena in their iterations. Boldt’s methods to construct Coxeter polynomials are reviewed, as well as A’Campo’s and Howlett’s Theorems.
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Barot, M., Jiménez González, J.A., de la Peña, JA. (2019). Concealedness and Weyl Groups. In: Quadratic Forms. Algebra and Applications, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-05627-8_4
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