Existence of the $de{t}^{S^2}$ map

Abstract

In this paper we show that for a vector space $V_d$ of dimension $d$ there exists a linear map $de{t}^{S^2}:V_d^{\otimes d(2d-1)}\to k$ with the property that $de{t}^{S^2}({\otimes }_{1\le i<j\le 2d}(v_{i,j}))=0$ if there exists $1\le x<y<z\le 2d$ 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 $de{t}^{S^2}$ to geometry and combinatorics.

12 pages, all comments are welcome