  We introduce new partial orders on the sequence positive roots and study the
statistics of the poset by using Auslander-Reiten quivers for finite type ADE.
Then we can prove that the statistics provide interesting information on the
representation theories of KLR-algebras, quantum groups and quantum affine
algebras including Dorey's rule, bases theory for quantum groups, and
denominator formulas between fundamental representations. As applications, we
prove Dorey's rule for quantum affine algebras $U_q(E_{6,7,8}^{(1)})$ and
partial information of denominator formulas for $U_q(E_{6,7,8}^{(1)})$. We also
suggest conjecture on complete denominator formulas for $U_q(E_{6,7,8}^{(1)})$.
