On the k-volume rigidity of a simplicial complex in ℝ ^d

We define a generic rigidity matroid for k-volumes of a simplicial complex in Rd and prove that for 2 ≤ k ≤ d − 1 it has the same rank as the classical generic d-rigidity matroid on the same vertex set (namely, the case k = 1). This is in contrast with the k = d case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of d-rigidity of the 1-skeleton of the complex and a combinatorial Hall condition on incidences of edges in k-faces.