J. A. Jimenez Gonzalez - A strong Gram classification of non-negative unit forms of Dynkin type A

fi:9330 - Fundamenta Informaticae, March 30, 2024, Volume 191, Issue 1
A strong Gram classification of non-negative unit forms of Dynkin type AArticle

Authors: J. A. Jimenez Gonzalez

    An integral quadratic form q is usually identified with a bilinear form b such that its Gram matrix with respect to the canonical basis is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type A_r and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49-82, 2021] and [Fundamenta Informaticae 185(3):221-246, 2022].

    Volume: Volume 191, Issue 1
    Published on: March 30, 2024
    Accepted on: February 17, 2024
    Submitted on: April 11, 2022
    Keywords: Mathematics - Combinatorics,15A63, 15A21, 15B36, 05C22, 05C50, 05C76, 05B20

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