Generic Rigidity of Graph Frameworks in Euclidean Space

Abstract

The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer and independently in 1970 by Laman, analogous edge-density counts on subgraphs fail in higher dimensions. In this paper, we solve the problem by providing a combinatorial characterization of generic infinitesimal rigidity valid in all dimensions. By gluing together local versions of Cramer's rule at each vertex, we construct a globally valid self-stress on the edges. The compatibility conditions governing these local solutions are controlled by the Pl\"ucker relations on the Grassmannian $Gr(d+1, v)$, allowing us to check generic rigidity using the combinatorics of Young's straightening law on tableaux.