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

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: 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 𝔸r (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.

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

DOI: 10.3233/FI-222109

Journal: Fundamenta Informaticae, vol. 185, no. 3, pp. 221-246, 2022