A strong Gram classification of non-negative unit forms of Dynkin type A

TL;DR

This paper characterizes strongly Gram congruent non-negative unit forms of Dynkin type A and completes their combinatorial classification using Coxeter polynomials.

Contribution

It proves the converse relation between Coxeter polynomial equality and strong Gram congruence for Dynkin type A forms, advancing their classification.

Findings

Converse holds for Dynkin type A_r with arbitrary corank.

Complete classification of non-negative unit forms of Dynkin type A.

Coxeter polynomial characterizes strong Gram congruence in this setting.

Abstract

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].