Jack Derangements

Abstract

For each integer partition $\lambda ⊢n$ we give a simple combinatorial expression for the sum of the Jack character ${\theta }_{\alpha }^{\lambda }$ over the integer partitions of $n$ with no singleton parts. For $\alpha =1,2$ this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. A byproduct of the latter is a simple combinatorial formula for the immanants of the matrix $J-I$ where $J$ is the all-ones matrix, which might be of independent interest. Our proofs center around a Jack analogue of a hook product related to Cayley's $\Omega$--process in classical invariant theory, which we call the principal lower hook product.