A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸n

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 𝔸n. 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 𝔸n. With these tools, connected principal unit forms of Dynkin type 𝔸n are also classified up to strong congruence.