Classification of polynomial dimension-product regimes

Characterize the triples of partitions (λ, μ, ν) of n for which f^λ ≥ f^μ ≥ f^ν implies that 1 ≤ f^μ f^ν / f^λ ≤ n^k for some fixed integer k.

Background

The quantum runtime for computing Kronecker coefficients depends on ratios or products of dimensions of the corresponding symmetric-group representations. The paper analyzes several dimension-growth regimes but does not fully identify all triples for which the relevant dimension expression is polynomially bounded.

A complete classification would identify the input regimes in which quantum algorithms could potentially have polynomial runtime while known classical methods remain nontrivial. The question is explicitly posed after the paper notes that cancellations in asymptotic dimension estimates make a general characterization difficult.