Polynomial-time computation of hook-case Kronecker coefficients

Establish a polynomial-time algorithm for computing the Kronecker coefficient g(λ, μ, ν) when ν = (n − a, 1^a) is a hook and λ and μ have Durfee square sizes at most a fixed constant k.

Background

The paper notes that Kronecker coefficients are polynomially bounded when all three partitions have fixed Durfee-square size, but exact computation is complicated by exponentially large alternating sums. It specifically identifies the hook case as unresolved even when λ and μ have bounded diagonal length. Existing combinatorial interpretations require constructing exponentially many tableaux, motivating the expectation that a more efficient algorithm should exist.