An integral quadratic form is called Coxeter-regular if its integer coefficients satisfy a divisibility condition equivalent to the fact that the associated Coxeter transformation and Weyl group are integral. Such forms are known to be useful in the study of finite-dimensional associative algebras, Lie algebras and certain singularities. We show that a non-negative (connected) Coxeter-regular form q is universal (that is, q represents all non-negative integers) if and only if q represents the integers 1,2,3,7 and 14. This may be viewed as a specialization (and, actually, an extension) of the Conway–Schneeberger/Bhargava “15 Theorem”. As one of the main tools we provide a complete classification, up to Z-equivalence, of all non-negative Coxeter-regular forms by means of so-called weak Dynkin type, which is a certain equivalence class of a Dynkin (bi)graph. In this way, we obtain a generalization of the known result of Barot-de la Peña for unit forms and simply-laced Dynkin diagrams.
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Nicolaus Copernicus University in Toruń, Toruń, Poland
Andrzej Mróz; Katarzyna Zając. Weak Dynkin type and the universality of non-negative Coxeter-regular integral quadratic forms. Documenta mathematica, Tome 30 (2025) no. 2, pp. 245-274. doi: 10.4171/dm/994
@article{10_4171_dm_994,
author = {Andrzej Mr\'oz and Katarzyna Zaj\k{a}c},
title = {Weak {Dynkin} type and the universality of non-negative {Coxeter-regular} integral quadratic forms},
journal = {Documenta mathematica},
pages = {245--274},
year = {2025},
volume = {30},
number = {2},
doi = {10.4171/dm/994},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/994/}
}
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AU - Katarzyna Zając
TI - Weak Dynkin type and the universality of non-negative Coxeter-regular integral quadratic forms
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