The Dynkin type of a non-negative unit form.
Let \(q: {\mathbb Z}^n\to{\mathbb Z}\) be an integral quadratic form given by the formula \[ q(x)=\sum_{j=1}^n q_jx_j^2+\sum_{i<j}q_{ij}x_ix_j. \] The form is said to be a unit form [resp. semi-unit form] if \(q_i=1\) [resp. \(q_i\in\{0,1\}\)] for \(i=1,\dots,n\). The main result of the paper asserts that for every connected non-negative semi-unit form \(q: {\mathbb Z}^n\to{\mathbb Z}\) the following statements hold. (a) There exists a \({\mathbb Z}\)-invertible transformation \(T: {\mathbb Z}^n \to {\mathbb Z}^n\) and a uniquely determined Dynkin diagram \(\Delta:= \text{Dyn}(q)\) (called the Dynkin type of \(q\)) such that \(q T(x_1,\dots, x_n) = q_{\Delta}(x_1,\dots, x_{n-c})\), where \(c\) is the corank of \(Q\) and \(q_{\Delta}\) is the quadratic form of the Dynkin diagram \({\Delta}\). (b) If \(q'\) is a connected restriction of \(q\), then \( \text{Dyn}(q')\leq \text{Dyn}(q)\). (c) There exists a connected restriction \(q'\) of \(q\) such that \(q'\) is a positive unit form and \( \text{Dyn}(q')= \text{Dyn}(q)\). It is also proved that a semi-unit form \(q\colon {\mathbb Z}^n\to{\mathbb Z}\) is positive if and only if \(q^{-1}(0)=\{0\}\) and \(-2\leq q_{ij}\leq2\) for all \(i<j\).
scientific article; zbMATH DE number 1394883