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

Abstract

With any symmetrizable integer Cartan matrix C∈SCarn⊆Mn(Z), a Z-invertible Coxeter matrix CoxC∈Mn(Z) is associated. We study such positive definite matrices up to a strong Gram Z-congruence C≈ZC′ (defined in the paper by means of a Z-invertible matrix B∈Mn(Z)) by means of the Dynkin type DynC, the complex spectrum speccC⊂C of the Coxeter polynomial coxC(t):=det⁡(tE−CoxC)∈Z[t] and the Coxeter type CtypC=(speccC,swC) of C. We show that the strong Gram Z-congruence C≈ZC′ implies the equality CtypC=CtypC′. 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., swC=swC′=1) and DynC is one of the simply laced Dynkin graphs An, n≥1, Dn, n≥4, E6, E7, E8, and (ii) n≥2, the matrices C,C′ are not symmetric and DynC is one of the Dynkin signed graphs Bn, Cn, F4, G2. 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 SCarn such that any positive definite irreducible symmetrizable integer Cartan matrix in Mn(Z), with n≤9, is strongly Gram Z-congruent with a matrix of the list, see Theorem 4.3.