  We introduce a path-theoretic framework for understanding the representation
theory of (quantum) symmetric and general linear groups and their higher level
generalisations over fields of arbitrary characteristic. Our first main result
is a "super-strong linkage principle" which provides degree-wise upper bounds
for graded decomposition numbers (this is new even in the case of symmetric
groups). Next, we generalise the notion of homomorphisms between Weyl/Specht
modules which are "generically" placed (within the associated alcove
geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally,
we provide evidence for a higher-level analogue of the classical Lusztig
conjecture over fields of sufficiently large characteristic.
