*Result*: One-peak posets with positive quadratic Tits form, their mesh translation quivers of roots, and programming in Maple and Python

*Abstract: By applying linear algebra and computer algebra tools we study finite posets with positive quadratic Tits form. Our study is motivated by applications of matrix representations of posets in representation theory, where a matrix representation of a partially ordered set , with a partial order , means a block matrix (over a field K) of size determined up to all elementary row transformations, elementary column transformations within each of the substrips , and additions of linear combinations of columns of to columns of , if . Drozd proves that T has only a finite number of direct-sum-indecomposable representations if and only if its quadratic Tits form is weakly positive (i.e., , for all non-zero vectors with integral non-negative coefficients). In this case, there exists an indecomposable representation M of size if and only if is a root of q, i.e., . Bondarenko and Stepochkina give a list of posets T with positive Tits form consisting of four infinite series and 108 posets, up to duality. In the paper, we construct this list in an alternative way by applying computational algorithms implemented in Maple and Python. Moreover, given any poset T of the list, we show that: (a) the Coxeter polynomial of the poset , obtained from T by adding a unique maximal element (called a peak), is a Coxeter polynomial of a simply-laced Dynkin diagram and (b) the set of -orbits of the set of integral roots of q admits a -mesh translation quiver structure of a cylinder shape, where is the Coxeter transformation of in the sense of Drozd . [Copyright &y& Elsevier]*