Path isomorphisms between quiver Hecke and diagrammatic Bott–Samelson endomorphism algebras

We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson's diagrammatic endomorphism algebras of Bott–Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche–Williamson's recent monograph and extend their categorical equivalence to cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza's categorical blob conjecture.