Output-sensitive computation of positive Littlewood–Richardson coefficients

Construct a classical algorithm that, for partitions λ, μ, and ν with |λ| = |μ| + |ν| and c^λ_{μν} > 0, computes c^λ_{μν} in time O(c^λ_{μν} poly(n)).

Background

Conjecture 4.7 strengthens the preceding dimension-product conjecture because the Littlewood–Richardson coefficient is bounded above by the relevant dimension product. The desired algorithm would be output-sensitive, with runtime governed by the coefficient itself. The paper emphasizes that the nontrivial cases are those in which the coefficient is polynomially large while the relevant partition lengths are not fixed.

References

We now pose a stronger conjecture which implies the previous one since cλ μν ≤ f λf μf ν .Conjecture 4.7. Let λ, μ, ν be partitions such that |λ| = |μ| + |ν|. Then there exists a classical algorithm running time O(cλ μν poly(n)) which computes cλ μν when cλ μ,ν > 0.

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Conjecture 4.7, Section 4, page 12