  In this paper we show that for a vector space $V_d$ of dimension $d$ there
exists a linear map $det^{S^2}:V_d^{\otimes d(2d-1)}\to k$ with the property
that $det^{S^2}(\otimes_{1\leq i<j\leq 2d}(v_{i,j}))=0$ if there exists $1\leq
x<y<z\leq 2d$ such that $v_{x,y}=v_{x,z}=v_{y,z}$. The existence of such a map
was conjectured in [4]. We present two applications of the map $det^{S^2}$ to
geometry and combinatorics.
