Derangements and tensor powers of adjoint modules for $\mathfrak{sl}_n$
Georgia Benkart and Stephen Doty

Abstract. We obtain the decomposition of the tensor space 
$\mathfrak{sl}_n^{\otimes k}$ as a module for $\mathfrak{sl}_n$,
find an explicit formula for the multiplicities of its 
irreducible summands, and (when $n \ge 2k$) describe the 
centralizer algebra $\mathcal C = \text{\rm End}_{\mathfrak{sl}_n}
(\mathfrak{sl}_n^{\otimes k})$ and its representations. The
multiplicities of the irreducible summands are derangement 
numbers in several important instances, and the dimension
of $\mathcal C$ is given by the number of derangements of a
set of $2k$ elements.