A horizontal mesh algorithm for posets with positive Tits form

Abstract

Following our paper [Fund. Inform. 136 (2015), 345--379],  we define a~horizontal mesh algorithm that constructs
a~$\widehat{\Phi}_I$-mesh translation quiver  $\Gamma(\widehat{\CR}_I,\widehat{\Phi}_I)$ consisting of
$\widehat{\Phi}_I$-orbits of the finite set $\widehat{\CR}_I=\{v\in\mathbb{Z}^I\; ;\;\widehat{q}_I(v)=1\}$ of Tits roots of
  a~poset $I$ with positive
definite Tits quadratic form  $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$.
Under the assumption that  $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$ is positive definite,  
the algorithm constructs $\Gamma(\widehat{\CR}_I,\widehat{\Phi}_I)$ such that it
is isomorphic with the $\widehat{\Phi}_D$-mesh translation quiver  
$\Gamma({\CR}_D,{\Phi}_D)$ of  $\widehat{\Phi}_D$-orbits of the finite set
${\CR}_D$ of roots  of a simply laced Dynkin quiver $D$ associated with $I$.