Six-Cycle Insertion and Hook-Unimodality in Higher Lie Characters

Let Mλ(x) denote the hook-multiplicity polynomial of the higher Lie character attached to a conjugacy class of cycle type λ. Adin–Hegedüs–Roichman conjectured that the coefficient sequence of Mλ(x) is unimodal for every partition λ. Using their product formula for rectangular cycle types, we prove the conjecture for the infinite family (6^s) for every s ≥ 1. In fact, (1+x)M_(6^s)(x) is strictly log-concave on its nonzero support. We then prove a six-cycle insertion theorem: adjoining any positive number of six-cycles preserves unimodality when the remaining cycle type is unimodal and contains no six. As a consequence, every nonempty partition all of whose parts lie in {1,2,3,4,5,6} has a unimodal hook-multiplicity sequence, with no bound on multiplicities or total size