Partitions of the complete hypergraph $$K_6^3$$ and a determinant-like function

Existence of the detS2$det^{S^2}$ map

“…Remark $$\Lambda _{V_d}^{S^3}$$ was introduced in [5] as another generalization of the exterior algebra. When $$d=2$$, it was shown that there exists a linear map $$det^{S^3}:V_2^{\otimes 20}\rightarrow k$$ with the property that $$det^{S^3}(\otimes _{1\leqslant i<j<k\leqslant 6}(v_{i,j,k}))=0$$ if there exist $$1<x<y<z<t\leqslant 6$$ such that $$v_{x,y,z}=v_{x,y,t}=v_{x,z,t}=v_{y,z,t}$$.…”