The Dynkin type of a non-negative unit form.
type: Item
instance of: scholarly article
title: The Dynkin type of a non-negative unit form.
zbMATH Open document ID: 1073.15531
author: M. Barot
author: José A. de la Peña
published in: Expositiones Mathematicae
publication date: 1999-01-01T00:00:00Z
review text: 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 \(i<j\).
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 1394883
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1394883