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

Abstract

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