Derangements and tensor powers of adjoint modules for \(\mathfrak{sl}_n\) (Q1862269)

The authors decompose the tensor space \(\mathfrak{sl}_n^{\otimes k}\) as a module for \(\mathfrak{sl}_n\) and give an explicit combinatorial formula for the multiplicities of its irreducible summands. For \(n\geq 2k\) the centralizer algebra \(\mathcal{C}=\text{End}_{\mathfrak{sl}_n} (\mathfrak{sl}_n^{\otimes k})\) is described via the walled Brauer algebra. The dimension of \(\mathcal{C}\) is computed combinatorially in terms of derangement numbers (here derangement is a permutation with no fixed elements). These numbers also describe the multiplicities of irreducible summands in \(\mathfrak{sl}_n^{\otimes k}\) mentioned above.