论文标题
Dynkin A型非负单元形式的强革分类
A strong Gram classification of non-negative unit forms of Dynkin type A
论文作者
论文摘要
通常用双线性形式B鉴定积分二次形式Q，以使其相对于规范基础的革兰氏矩阵是上三角形的。如果它们相应的上三角双线性形式（分别对称对称性）等效，则两种整体二次形式被强烈（弱）的革兰氏酶表示一致。如果Q是统一的，则这种上三角双线性形式是单模型的，并且人们考虑了相关的Coxeter转换及其特征多项式，即Q的所谓的Coxeter多项式，并具有此识别。两种强gram一致的二次单元形式是弱革兰氏剂的一致性，并且具有相同的Coxeter多项式。在这里，我们表明，此陈述的相反是针对Dynkin型A_R和任意Corank的连接的非阴性案例，并使用此特征来完成此类二次形式的组合分类，该形式始于[FuldActionala Informaticae 184（1）：49-82，2021]和[Fundementa Informaticae 185（Fundementa Informaticae）185（3）（3）：221-246，221-24，221-24，221-24，221，246，221-24，221-24，221，246，221，246，221-24，221-24，221，246，221，24，202222。
An integral quadratic form q is usually identified with a bilinear form b such that its Gram matrix with respect to the canonical basis is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type A_r and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49-82, 2021] and [Fundamenta Informaticae 185(3):221-246, 2022].
