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

TL;DR

This paper introduces a determinant-like map $det^{S^3}$ for hypergraphs, exploring its properties and providing an explicit formula, thus extending algebraic tools to hypergraph partitions.

Contribution

The paper defines a new determinant-like function $det^{S^3}$ and constructs a graded vector space $ ext{Lambda}^{S^3}_V$ with properties analogous to classical algebraic structures.

Findings

Existence and uniqueness of $det^{S^3}$ established.

Explicit formula for $det^{S^3}$ as a sum over 2-partitions.

Dimension result for $ ext{Lambda}^{S^3}_{V_2}[6]$.

Abstract

In this paper we introduce a determinant-like map $det^{S^3}$ 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 \cite{sta2}. When $dim(V_2)=2$ we show that $dim_k(\Lambda^{S^3}_{V_2}[6])=1$ which gives the existence and uniqueness of $det^{S^3}$. We also give an explicit formula for $det^{S^3}$ as a sum over certain $2$-partitions of the complete hypergraph $K_6^3$.