Symmetric factorizations of the complete uniform hypergraph

Abstract

A factorization of the complete k-hypergraph $(V,V^{\{k\}})$ of index $s\ge 2$, simply a (k, s) factorization on V, is a partition $\{F_1,F_2,\ldots , F_s\}$ of the edge set $V^{\{k\}}$ into s disjoint subsets such that each k-hypergraph $(V,F_i)$, called a factor, is a spanning subhypergraph of $(V,V^{\{k\}})$. A (k, s) factorization $\{F_1,F_2,\ldots , F_s\}$ on V is symmetric if there is a subgroup G of the symmetric group $\mathrm{Sym}(V)$ such that G induces a transitive action on $\{F_1,F_2,\ldots , F_s\}$ and for each i, the stabilizer $G_{F_i}$ is transitive on both V and $F_i$. A symmetric factorization on V is homogeneous if all its factors admit a common transitive subgroup of $\mathrm{Sym}(V)$. In this paper, we give a complete classification of symmetric (k, s) factorizations on a set of size n under the assumption that $s\ge 2$ and $6\le 2k\le n$. It is proved that, up to isomorphism, there are two infinite families and 29 sporadic examples of symmetric factorizations which are not homogeneous. Among these symmetric factorizations, only eight of them are not 1-factorizations.