Reduction of Weakly Definite Unit Forms

Abstract

In the following we consider integral quadratic forms q : ℤⁿ → ℤ of the form $q(x) = \sum\nolimits_{i} {x_{i}^{2}} + \sum\nolimits_{{i < j}} {{q_{{ij}}}{x_{i}}{x_{j}}}$, called unit forms. Such unit forms appear naturally (as Tits— or Euler forms) in various areas of representation theory in connection with the characterization of the representation type and the examination of the Jordan-Hölder multiplicities of the indecomposable representations (see for instance [11, 1, 10]). In this context the main problems concerning a given unit form q are the following. 1) Decide whether or not is weakly non-negative (resp. weakly positive), that is, q satisfies q(x) ≥ 0 (resp. q(x) > 0) for all 0 ≠ x ∈ ℕn 2) Describe the set ${\mathcal{O}_{q}} = \left\{ {0 \ne x \in {\mathbb{N}^{n}}|q(x) = 0} \right\}$ of positive zero roots if q is weakly non-negative and the set ${\mathcal{R}_{q}} = \left\{ {x \in {\mathbb{N}^{n}}|q(x) = 1} \right\}$ of positive roots if q is weakly positive. There are several concepts to approach these problems; a survey is given by the author in [10].