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Quadratic Forms

Combinatorics and Numerical Results

Authors: Dr. Michael Barot, Prof. Dr. Jesús Arturo Jiménez González, Prof. Dr. José-Antonio de la Peña

Publisher: Springer International Publishing

Book Series : Algebra and Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This monograph presents combinatorial and numerical issues on integral quadratic forms as originally obtained in the context of representation theory of algebras and derived categories.

Some of these beautiful results remain practically unknown to students and scholars, and are scattered in papers written between 1970 and the present day. Besides the many classical results, the book also encompasses a few new results and generalizations.

The material presented will appeal to a wide group of researchers (in representation theory of algebras, Lie theory, number theory and graph theory) and, due to its accessible nature and the many exercises provided, also to undergraduate and graduate students with a solid foundation in linear algebra and some familiarity on graph theory.

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Table of Contents

Frontmatter

Chapter 1. Fundamental Concepts

Abstract

Throughout the chapter we introduce important concepts as well as basic results: given an integral quadratic form \(q:\mathbb {Z}^n \to \mathbb {Z}\), numbers of the form q(v) for a vector v in \(\mathbb {Z}^n\) are said to be represented by q, and the form q is said to be universal if every positive integer is represented by q. We sketch the proof of Conway and Schneeberger’s Fifteen Theorem, which states that a positive integral form with associated symmetric matrix having integer coefficients is universal if and only if it represents all positive integers up to 15. We also survey the theory of binary integral quadratic forms (due originally to Gauss), and apply it to specific examples referred to as Kronecker and Pell forms. Finally, we find general conditions for a real quadratic form \(q_{\mathbb {R}}:\mathbb {R}^n \to \mathbb {R}\) to be positive (that is, \(q_{\mathbb {R}}(x)>0\) for any nonzero vector x) or nonnegative (\(q_{\mathbb {R}}(x)\geq 0\) for any vector x in \(\mathbb {R}^n\)).

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Chapter 2. Positive Quadratic Forms

Abstract

An integral quadratic form is a unit form if all its diagonal coefficients are equal to one. In this chapter we study positive unit forms, that is, those integral quadratic unit forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) with q(x) > 0 for any nonzero vector x in \(\mathbb {Z}^n\). A unit form q is critical nonpositive if it is not positive, but each proper restriction of q is. A vector v is called radical for q if q(v + u) = q(u) for any vector u in \(\mathbb {Z}^n\). We prove Ovsienko’s Criterion: a unit form in n ≥ 3 variables is critical nonpositive if and only if q is nonnegative with radical generated by a radical vector with no zero among its entries. One of the most important tools in the theory of integral quadratic forms, inflations and deflations, are introduced in this chapter, and are used to provide a classification of positive unit forms in terms of Dynkin types. A combinatorial characterization of such forms in terms of assemblers of graphs is also presented.

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Chapter 3. Nonnegative Quadratic Forms

Abstract

Let G be a connected graph. In this Chapter we show that q _G is a nonpositive, nonnegative unit form if and only if G is an extended Dynkin diagram, and give a short proof for Vinberg’s characterization of such diagrams. As shown by Barot and de la Peña, for a nonnegative semi-unit form q there is an iterated flation T such that qT = q _Δ ⊕ ξ ^c, where Δ is a disjoint union of Dynkin diagrams and ξ ^c is the zero quadratic form in c variables. The uniquely determined union of Dynkin diagrams Δ is referred to as the Dynkin type of q, while c is the corank of q (the rank of the radical of q). Hypercritical nonnegative unit forms are considered (those borderline forms between positive and nonnegative forms), and a characterization of such forms is provided. We say that two unit forms q and q′ are root equivalent if q is a q′-root induced form, and q′ is a q-root induced form. Here we show that two non-negative semi-unit forms have the same Dynkin type if and only if they are root equivalent, and derive an interesting partial order in the set of Dynkin types.

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Chapter 4. Concealedness and Weyl Groups

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.

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Chapter 5. Weakly Positive Quadratic Forms

Abstract

In this chapter we analyze integral quadratic forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) satisfying q(x) > 0 for any positive vector x in \(\mathbb {Z}^n\), so-called weakly positive unit forms, and their sets of positive roots R ⁺(q). A characterization of critical unit forms, those forms that fail to be weakly positive but all their restrictions are, is presented. We give criteria to identify weakly positive forms, for instance finiteness of the set of positive roots R ⁺(q) (due to Drozd and Happel), and Zeldych’s Theorem (looking for properties of principal submatrices of the symmetric matrix associated to q). Ovsienko’s Theorem is also proved, setting 6 as a bound for the entries of all positive roots of a weakly positive unit form. Those forms with a positive root reaching bound 6 are studied, following Ostermann and Pott. A classification of thin forms due to Dräxler, Drozd, Golovachtchuk, Ovsienko and Zeldych is sketched, as well as a procedure to find all weakly positive unit forms starting with (good) thin forms.

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Chapter 6. Weakly Nonnegative Quadratic Forms

Abstract

In this chapter we analyze integral quadratic forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) satisfying q(x) ≥ 0 for any positive vector x in \(\mathbb {Z}^n\), so-called weakly nonnegative semi-unit forms. Here a prominent role is played by maximal and locally maximal positive roots of q, which can be used to characterize weak nonnegativity. We also describe hypercritical semi-unit forms, those forms not weakly nonnegative such that any proper restriction is weakly nonnegative. Diverse criteria for weak nonnegativity are provided, including Zeldych’s Theorem and a few algorithms using iterated edge reductions, following von Höhne and de la Peña. A generalization of Ovsienko’s Theorem due to Dräxler, Golovachtchuk, Ovsienko and de la Peña is proved in the last section, for which Ringel’s concepts of graphical and semi-graphical forms are essential.

Michael Barot, Jesús Arturo Jiménez González, José-Antonio de la Peña

Backmatter

Title: Quadratic Forms
Authors: Dr. Michael Barot
Prof. Dr. Jesús Arturo Jiménez González
Prof. Dr. José-Antonio de la Peña
Copyright Year: 2019
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-05627-8
Print ISBN: 978-3-030-05626-1
DOI: https://doi.org/10.1007/978-3-030-05627-8

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