All Kronecker coefficients are reduced Kronecker coefficients (Q6642431)

The Kronecker coefficient \(\mathbf{k}(\lambda, \mu, \nu)\) of the symmetric group \(S_{n}\) is the multiplicity of the irreducible \(S_{n}\) representation \(\mathbb{S}_{\nu}\) in the tensor product \(\mathbb{S}_{\lambda} \otimes \mathbb{S}_{\mu}\). The reduced Kronecker coefficients \(\overline{\mathbf{k}}(\alpha, \beta, \gamma)\) are defined as the stable limit of the ordinary Kronecker coefficients. These coefficients are called extended Littlewood-Richardson numbers in [\textit{A. N. Kirillov}, Publ. Res. Inst. Math. Sci. 40, No. 4, 1147--1239 (2004; Zbl 1077.05098)].\N\NIn the paper under review the authors settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. Moreover, as a corollary, they deduce that deciding the positivity of reduced Kronecker coefficients is \(\mathsf{NP}\)-hard, and computing them is \(\mathsf{\#P}\)-hard under parsimonious many-one reductions. The proof provides an explicit isomorphism of the corresponding highest weight vector spaces.