A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices

With any symmetrizable integer Cartan matrix C is an element of SCar(n) subset of M-n(Z), a Z-invertible Coxeter matrix Cox(C) is an element of M-n(Z) is associated. We study such positive definite matrices up to a strong Gram Z-congruence C approximate to(Z) C' (defined in the paper by means of a 7G-invertible matrix B is an element of M-n(Z)) by means of the Dynkin type Dyne, the complex spectrum specc(C) subset of C of the Coxeter polynomial cox(C) (t) := det(tE - Cox(C)) is an element of Z[t] and the Coxeter type CtYP(C) = (specc(C),sw(C)) of C. We show that the strong Gram Z-congruence C approximate to(z) C' implies the equality Ctyp(C) = Ctyp(C)'. The inverse implication is an open problem studied in the paper. However, we prove the implication for positive definite symmetrizable integer Cartan matrices C, C' satisfying any of the following two conditions: (i) if n <= 9, the matrices C, C' are symmetric (i.e., sw(C) = sw(C), = 1) and Dyne is one of the simply laced Dynkin graphs A(n), n >= 1, D-n, n >= 4, E-6, E-7, E-8, and (ii) n >= 2, the matrices C, C' are not symmetric and Dyn(C) is one of the Dynkin signed graphs B-n, C-n, F-4, G(2). We do it by a reduction to an analogous problem for positive Cox-regular edge-bipartite graphs. Moreover, given n <= 9, we present a list of 60 pairwise non-congruent matrices in SCar(n) such that any positive definite irreducible symmetrizable integer Cartan matrix in M-n(Z), with n <= 9, is strongly Gram Z-congruent with a matrix of the list, see Theorem 4.3. (C) 2019 Elsevier Inc. All rights reserved.