Positive Reflection Symmetrizers: Random Walks and an Abel–Young Character of Ordered Trees

Let W be a finite well-generated complex reflection group. We introduce
an ordered reflection symmetrizer obtained by averaging products of cyclic
Reynolds projectors over rank-sized reflection generating sets, and prove that
it is central and positive semidefinite in every unitary representation via a re
cursion over codimension-one parabolic subgroups. In type A, the normalized
symmetrizer is a random walk generated by independently activating edges of
a uniformly edge-ordered Cayley tree. We determine its exact class law and an
exact convex-mixture representation by conjugacy-averaged uniform measures
on Young subgroups; the multiplicities form a nonnegative integral Young
permutation character. Abel real-rootedness and normalized Schur mono
tonicity order the complete Specht spectrum by dominance and give the exact
spectral gap. We also prove total-variation cutoff at logk/(log2 + 1/2). For
G(m,1,2), we obtain an exact subgroup-idempotent formula and normalized
spectrum {0,1/4,1}.