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 T={p1,…,pn}, with a partial order ⪯, means a block matrix M=[M1|M2|…|Mn] (over a field K) of size d*×(d1,…,dn) determined up to all elementary row transformations, elementary column transformations within each of the substrips M1,M2,…,Mn, and additions of linear combinations of columns of Mi to columns of Mj, if pi≺pj. Drozd [10] proves that T has only a finite number of direct-sum-indecomposable representations if and only if its quadratic Tits form q(x1,…,xn,x*)=x12+⋯+xn2+x*2+∑pi≺pjxixj-x*(x1+⋯+xn) is weakly positive (i.e., q(a1,…,an,a*)>0, for all non-zero vectors (a1,…,an,a*) with integral non-negative coefficients). In this case, there exists an indecomposable representation M of size d*×(d1,…,dn) if and only if (d1,…,dn,d*) is a root of q, i.e., q(d1,…,dn,d*)=1. Bondarenko and Stepochkina [8] 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 coxI(t) of the poset I=T∪{*}, obtained from T by adding a unique maximal element * (called a peak), is a Coxeter polynomial FΔI(t) of a simply-laced Dynkin diagram ΔI∈{An,Dn,E6,E7,E8}and (b) the set of Φ-orbits of the set Rq of integral roots of q admits a Φ-mesh translation quiver structure Γ(Rq,Φ) of a cylinder shape, where Φ is the Coxeter transformation of I=T∪{*} in the sense of Drozd [10].