Coxeter invariants for non-negative unit forms of Dynkin type A

TL;DR

This paper introduces a combinatorial invariant for non-negative unit forms of Dynkin type A, enabling the determination of Coxeter polynomials and numbers, thus advancing the classification of these algebraic structures.

Contribution

It provides a new combinatorial strong Gram invariant for non-negative Dynkin type A unit forms, extending previous frameworks and facilitating explicit calculations of Coxeter invariants.

Findings

Identified all Coxeter polynomials for the class

Computed Coxeter numbers explicitly

Established a classification framework for these forms

Abstract

Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are Z-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type A, within the framework introduced in [Fundamenta Informaticae 184(1):49-82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.