Decompositions of Complete Multipartite Graphs into Complete Graphs

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of $K_\ell$ in $G(k,n)$ is a subgraph of exactly one copy of $K_k$ in $X$. This paper asks: when does $G(k,n)$ have an $\ell$-decomposition? The answer is well known for the $\ell=2$ case. In particular, $G(k,n)$ has a 2-decomposition if and only if there exists $k-2$ mutually orthogonal Latin squares of order $n$. For general $\ell$, we prove that $G(k,n)$ has an $\ell$-decomposition if and only if there are $k-\ell$ Latin cubes of dimension $\ell$ and order $n$, with an additional property that we call mutually invertible. This property is stronger than being mutually orthogonal. An $\ell$-decomposition of $G(k,n)$ is then constructed whenever no prime less than $k$ divides $n$.