Existence of the Map $de{t}^{S^3}$

Abstract

In this paper we show the existence of a nontrivial linear map $de{t}^{S^3}:V_d^{\otimes \left(\genfrac{}{}{0ex}{}{3d}{3}\right)}\to k$ with the property that $de{t}^{S^3}({\otimes }_{1\le i<j<k\le 3d}(v_{i,j,k}))=0$ if there exists $1\le x<y<z<t\le 3d$ 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 $de{t}^{S^3}$ to study those d-partitions of the complete hypergraph $K_{3d}^3$ that have zero Betti numbers. We also discuss algebraic and combinatorial properties of a map $de{t}^{S^r}:V_d^{\otimes \left(\genfrac{}{}{0ex}{}{rd}{r}\right)}\to k$ which generalizes the determinant map, the map $de{t}^{S^2}$ from [9], and $de{t}^{S^3}$.

27 pages, comments welcome, update to previous paper on ArXiV with one section and one subsection added (applications to combinatorics)