Structure and a Coxeter–Dynkin type classification of corank two non-negative posets

Inspired by famous results of Drozd (1974) [13], Nazarova–Roĭter (1975) [42], and Bondarenko–Stepochkina (2005) [8], on the importance of finite posets I   with the non-negative Tits quadratic form qˆI:ZI→Z in the representation theory, we study the finite non-negative posets I of corank two, that is, such posets I   that the symmetric Gram matrix GI:=12[CI+CItr]∈MI(Q) of I   is positive semi-definite of corank two, where CI∈MI(Z)CI∈MI(Z) is the incidence matrix of I. A structure of such posets is described in the paper. It is shown that every such a poset I can be obtained in a natural way from a corank zero poset J and a pair of roots of J. Following main ideas of the Coxeter spectral analysis of posets, we associate with any connected corank two poset I   the incidence quadratic form qI:ZI→ZqI:ZI→Z, the Coxeter matrix CoxI:=−CI⋅CI−tr, the Coxeter polynomial coxI(t):=det⁡(t⋅E−CoxI)∈Z[t]coxI(t):=det⁡(t⋅E−CoxI)∈Z[t], its complex Coxeter spectrum speccIspeccI, the Coxeter number cI∈N∪{∞}cI∈N∪{∞}, the reduced Coxeter number cˇI∈N, the incidence defect homomorphism ∂˜I:ZI→KerqI⊆ZI, and a simply laced Dynkin diagram DynIDynI. In particular, we study the question when the Coxeter spectrum speccIspeccI of I  , together with DynIDynI, determines the incidence matrix CICI of I (hence the poset I  ) uniquely, up to a ZZ-congruence.By applying linear algebra techniques and a combinatorial computer-aided analysis of posets, we prove that the question has an affirmative answer for connected corank two posets I  , with at most 15 elements. A complete list of such posets is presented and their Coxeter type invariants are computed. It follows that |I|≥6|I|≥6, there is only one such a poset I   of cardinality 6, and 8 such posets of cardinality 7, up to the poset duality I↦IopI↦Iop. The existence of a ΦIΦI-mesh translation quiver Γ(RI∪KerqI,ΦI) on the set RI∪KerqI is discussed, where RI={v∈ZI;qI(v)=1} is the set of roots of qIqI.