On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams

We continue our Coxeter type study of a class of finite posets we started in Gąsiorek and Simson (2012). Here we present a more general algorithmic approach to a classification problem for arbitrary posets JJ, with n≥2n≥2 elements, that are positive (resp. non-negative) in the sense that the symmetric matrix CJ+CJtr∈Mn(Z) is positive definite (resp. positive semi-definite), where CJ∈Mn(Z)CJ∈Mn(Z) is the incidence matrix of JJ. In particular we show that the Coxeter spectral classification of positive (resp. non-negative) posets can be effectively solved using the right action ∗:Mn(Z)×Gl(n,Z)D→Mn(Z), A↦A∗B≔Btr⋅A⋅BA↦A∗B≔Btr⋅A⋅B, of isotropy groups Gl(n,Z)D of simply laced Dynkin (resp. Euclidean) diagrams DD. By applying recent results of the second author in [SIAM J. Discrete Math. 27 (2013) 827–854] we show that, given two connected positive posets II and JJ with at most 1010 points: (i) the incidence matrices CICI and CJCJ of II and JJ are ZZ-congruent if and only if the Coxeter spectra of II and JJ coincide, and (ii) the matrix CICI is ZZ-congruent with its transpose CItr. Analogous results for non-negative posets of corank one or two are also discussed.