On the relationship between rank-$(n-1)$ convexity and $\mathcal{S}$-quasiconvexity

We prove that rank-$(n-1)$ convexity does not imply ${\mathcal S}$-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in ${\mathbb{M}}^{m\times n}$ for $m>n$, by adapting the well-known Sverak’s counterexample [5] to the solenoidal setting. On the other hand, we also remark that rank-$(n-1)$ convexity and $\mathcal{S}$-quasiconvexity turn out to be equivalent in the space of $n\times n$ diagonal matrices. This follows by a generalization of Mueller’s work [4].