An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$

TL;DR

This paper introduces a new acyclic $d$-partition of the $r$-uniform complete hypergraph, proving its homogeneity and acyclicity, and applies this to show the nontriviality of a specific map, addressing a conjecture.

Contribution

It presents a novel acyclic $d$-partition of the hypergraph and demonstrates its properties, providing insights into a conjecture about the map $det^{S^r}$.

Findings

The $d$-partition $ E_d^{(r)}$ is homogeneous.

Each part $ O_i^{(r,d)}$ is acyclic.

The map $det^{S^r}$ is nontrivial for all $r$.

Abstract

In this paper we introduce a $d$-partition $\mathcal{E}_d^{(r)}=(\Omega_1^{(r,d)}, \Omega_2^{(r,d)},\dots, \Omega_d^{(r,d)})$ of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$. We prove that $\mathcal{E}_d^{(r)}$ is homogeneous and that each hypergraph $\Omega_i^{(r,d)}$ is acyclic (i.e. has zero Betti numbers). As an application, we show that the map $det^{S^r}$ is nontrivial for every $r$, which gives a partial answer to a conjecture from [14].