Characterization of polynomial dimension-product regimes

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

Background

The paper studies when quantum runtime bounds expressed through dimensions of symmetric-group representations are polynomial or superpolynomial. For Kronecker coefficients, positivity implies a dimension ratio bounded below by one, while the relevant upper ratio can vary substantially across partition shapes. The paper gives examples of regimes where the ratio is polynomial, but does not provide a complete characterization, especially when leading asymptotic terms cancel.

References

However, it is not clear how to characterize all regimes considered in [LH24], as the dimensions can have polynomial, exponential and superexponential growths, but in the considered ratios the leading terms could cancel. Question 2. Characterize the triples of partitions (λ, μ, ν) of n, such that if f λ ≥ f μ ≥ f ν then 1 ≤ f μf νf λ ≤ nk for some fixed integer k.

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Question 2, Section 1, page 3

Question 2. Characterize the triples of partitions (λ, μ, ν) of n, such that if f λ ≥ f μ ≥ f ν then 1 ≤ f μf νf λ ≤ nk for some fixed integer k.

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Question 2, Section 1, page 3