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

Bondarenko, V. M., Styopochkina, M. V. (2018). Strengthening of a theorem on Coxeter–Euclidean type of principal partially ordered sets. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics(4), 8–15. https://doi.org/10.17721/1812-5409.2018/4.1

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.Key words: positive and non-negative quadratic form, quadratic Tits form, P-critical poset, principal poset, Dynkin diagram, Euclidean diagram.Pages of the article in the issue: 8 - 15Language of the article: English