A graph theoretical framework for the strong Gram classification of non-negative unit forms of Dynkin type A

TL;DR

This paper develops a graph theoretical framework to analyze strong Gram classification of non-negative unit forms of Dynkin type A, introducing new techniques and classifications for these algebraic structures.

Contribution

It introduces a modified inflation technique and the concept of inverse quivers to study strong Gram congruence and classifies connected principal forms of Dynkin type A.

Findings

Weak and strong Gram congruence coincide for positive unit forms of Dynkin type A.

A new method to analyze the Coxeter matrix of these forms is proposed.

Connected principal forms of Dynkin type A are fully classified.

Abstract

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type A. The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type A. Connected principal unit forms of such type are also classified.