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

Abstract

In this paper we introduce a determinant-like map $de{t}^{S^3}$ and study some of its properties. For this we define a graded vector space ${\Lambda }_V^{S^3}$ that has similar properties with the exterior algebra ${\Lambda }_V$ and the exterior GSC-operad ${\Lambda }_V^{S^2}$ from \cite{sta2}. When $dim(V_2)=2$ we show that $di{m}_k({\Lambda }_{V_2}^{S^3}[6])=1$ which gives the existence and uniqueness of $de{t}^{S^3}$. We also give an explicit formula for $de{t}^{S^3}$ as a sum over certain $2$-partitions of the complete hypergraph $K_6^3$.

27 pages, 25 figures, all comments are welcome