Dimension-product-time algorithm for Littlewood–Richardson coefficients

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

Background

Littlewood–Richardson coefficients count tableaux and can also be represented as integer-point counts in hive polytopes. The paper obtains efficient algorithms in several restricted regimes, including fixed-length cases, but the general dimension-product runtime remains unresolved. Conjecture 4.6 restates a conjecture from Larocca and Havlíček and seeks a classical algorithm matching the relevant quantum runtime up to polynomial factors.

References

In the same spirit as with the Kostka coefficients we reiterate the conjecture of [LH24]. Conjecture 4.6 ([LH24]). Let λ, μ, ν be partitions such that |λ| = |μ| + |ν|. Then there exists a classical algorithm running in time O( f λf μf ν poly(n)) which computes cλ μν.

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Conjecture 4.6, Section 4, page 12

Conjecture 4.6 ([LH24]). Let λ, μ, ν be partitions such that |λ| = |μ| + |ν|. Then there exists a classical algorithm running in time O( f λf μf ν poly(n)) which computes cλ μν.

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Conjecture 4.6, Section 4, page 12