On the k-volume rigidity of a simplicial complex in Rᵈ

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.