Jack Derangements

2023/04/13 by Lindzey, Nathan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2304.06629

Abstract

For each integer partition λ\vdash n we give a simple combinatorial expression for the sum of the Jack character θ^λ_α over the integer partitions of n with no singleton parts. For α= 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 Ω--process in classical invariant theory, which we call the principal lower hook product.

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• On the second largest eigenvalue of certain graphs in the perfect matching association scheme

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