  We recast the classical notion of standard tableaux in an alcove-geometric
setting and extend these classical ideas to all reduced paths in our geometry.
This broader path-perspective is essential for implementing the higher
categorical ideas of Elias--Williamson in the setting of quiver Hecke algebras.
Our first main result is the construction of light leaves bases of quiver Hecke
algebras. These bases are richer and encode more structural information than
their classical counterparts, even in the case of the symmetric groups. Our
second main result provides path-theoretic generators for the Bott--Samelson
truncation of the quiver Hecke algebra.
