  In this paper, we define a number of closely related isomorphisms. On one
side of these isomorphisms sit a number of of algebras generalizing the Hecke
and affine Hecke algebras, which we call the "Hecke family"; on the other, we
find generalizations of KLR algebras in finite and affine type A, the "KLR
family."
  We show that these algebras have compatible isomorphisms generalizing those
between Hecke and KLR algebras given by Brundan and Kleshchev. This allows us
to organize a long list of algebras and categories into a single system,
including (affine/cyclotomic) Hecke algebras, (affine/cyclotomic) $q$-Schur
algebras, (weighted) KLR algebras, category $\mathcal{O}$ for $\mathfrak{gl}_N$
and for the Cherednik algebras for the groups $\mathbb{Z}/e\mathbb{Z}\wr S_m$,
and give graded presentations of all of these objects.
