Generic rigidity and the Grassmannian

We characterize the generic rigidity of bar-joint frameworks in arbitrary dimensions. To bypass the failure of subgraph edge-sparsity counts in $d > 2$, we analyze the existence of self-stresses by applying Cramer's rule locally at each vertex. We encode this local data as weights on a directed acyclic orientation of the framework's line graph. The resulting explicit compatibility conditions can be checked using Pl\"ucker relations on the Grassmannian, reducing the generic rigidity problem to the combinatorics of standard tableaux and Young's straightening law.