Existence of the Map $det^{S^3}$

In this paper we show the existence of a nontrivial linear map $det^{S^3}:V_d^{\otimes\binom{3d}{3}}\to k$ with the property that $det^{S^3}(\otimes_{1\leq i<j<k\leq 3d}(v_{i,j,k}))=0$ if there exists $1\leq x<y<z<t\leq 3d$ such that $v_{x,y,z}=v_{x,y,t}=v_{x,z,t}=v_{y,z,t}$. This gives a partial answer to a conjecture from [10]. As an application, we use the map $det^{S^3}$ to study those d-partitions of the complete hypergraph $K^3_{3d}$ that have zero Betti numbers. We also discuss algebraic and combinatorial properties of a map $det^{S^r}:V_d^{\otimes\binom{rd}{r}}\to k$ which generalizes the determinant map, the map $det^{S^2}$ from [9], and $det^{S^3}$.