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

We construct an explicit isomorphism between certain 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 equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. In the special case of the symmetric group this gives an elementary and more explicit proof of the tilting character formula of Riche-Williamson's recent monograph.