Strengthening of a theorem on Coxeter–Euclidean type of principal partially ordered sets

Abstract

Among the quadratic forms, playing an important role in modern mathematics, the Tits quadratic forms should be distinguished. Such quadratic forms were first introduced by P. Gabriel for any quiver in connection with the study of representations of quivers (also introduced by him). P. Gabriel proved that the connected quivers with positive Tits form coincide with the Dynkin quivers. This quadratic form is naturally generalized to a poset. The posets with positive quadratic Tits form (analogs of the Dynkin diagrams) were classified by the authors together with the P-critical posets (the smallest posets with non-positive quadratic Tits form). The quadratic Tits form of a P-critical poset is non-negative and corank of its symmetric matrix is 1. In this paper we study all posets with such two properties, which are called principal, related to equivalence of their quadratic Tits forms and those of Euclidean diagrams. In particular, one problem posted in 2014 is solved.

On classifying the non-Tits P-critical posets

“…In this section we write out a table (Table 2) of all Tits P -critical posets up to isomorphism and duality, i.e. Table 1 [7] (see also [16] ) of the P -critical posets without the posets from Table 1 of Section 1.…”