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

Article type: Research Article

Authors: Jiménez González, Jesús Arturo^{; *}

Affiliations: Instituto de Matemáticas, UNAM, Mexico. jejim@im.unam.mx

Correspondence: [*] Address of correspondence: Instituto de Matemáticas, Mexico City, Mexico.

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.

Keywords: Integral quadratic form, Gram congruence, quiver, Dynkin type, Coxeter matrix, edge-bipartite graph, signed line graph, 2020 MSC: 15A63, 15A21, 15B36, 05C22, 05C50, 05C76, 05B20

DOI: 10.3233/FI-2021-2092

Journal: Fundamenta Informaticae, vol. 184, no. 1, pp. 49-82, 2021