  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 $n$.
