Regular patterns in Coxeter groups
The proof of Brink & Howlett uses minimal (elementary) roots.
  • One positive root a dominates another b when the region b > 0 in the Tits cone is contained in a > 0. A root is minimal if it dominates only itself. Bédard's group again.
  • The number of minimal roots is finite (Brink & Howlett).
  • When the group is finite, all roots a are minimal. When affine, minimal roots are the a > 0 and -a + 1 with a for the finite system.