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

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.