The study of the canonical and dual canonical bases of quantum groups was one of the
main motivations for the introduction of cluster algebras by Fomin and Zelevinsky around
2000. Their observations had led them to conjecture that cluster monomials of quantum
coordinate rings were elements of the dual canonical basis. This conjecture was recently proved
by Kang-Kashiwara-Kim-Oh using certain categorifications of quantum groups compatible
with their cluster structure. The key tool is the construction of non-trivial R-matrices in
categories of modules over certain algebras called quiver Hecke algebras (or KLR algebras)
introduced by Khovanov-Lauda and Rouquier around 2010. We will explain how the results of
Kang-Kashiwara-Kim-Oh yield remarkable compatibilities between various partial orderings
of different natures. We will show that these compatibilities imply certain combinatorial
relationships between multisegments parametrising irreducible representations of quiver Hecke
algebras and certain gradings of cluster monomials called g-vectors. We will also relate this
to a recent work of Kashiwara-Kim (2018).