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Coxeter Invariants for Non-negative Unit Forms of Dynkin Type A

Jiménez González Jesús Arturo

Fundamenta Informaticae

2022

Vol. 185, nr 3

221--246

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 r (for r ≥ 1), 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.

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

González Jiménez, Arturo Jesús

Fundamenta Informaticae

2021

Vol. 184, nr 1

49--82

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.