The ultimate goal of representation theory is to obtain a complete understanding of the submodule structure of some algebraic objects. In this talk we will tackle this problem for KLR algebras. In particular, we will discover the rich algebraic structure of KLR algebras through an exciting interplay with the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. We will reveal how combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Even further we go beyond just looking at sets of Dyck tableaux (which enumerate the q-decomposition numbers) — we uncover the relationships for passing between them. This "meta-Kazhdan-Lusztig combinatorics" is, in fact, rich enough to fully determine the Ext-quiver and the relations presentation of these algebras.