Combinatorial description of Kronecker coefficients

Determine a combinatorial description of the Kronecker coefficients \(g_{\lambda,\mu}^{\nu}\), which give the multiplicities of irreducible representations \(S_\nu\) in tensor products \(S_\lambda\otimes S_\mu\) of irreducible representations of symmetric groups.

Background

Kronecker coefficients are structural constants in the representation theory of symmetric groups. For partitions λ,μ,ν\lambda,\mu,\nu of nn, the coefficient gλ,μνg_{\lambda,\mu}^{\nu} is the multiplicity of SνS_\nu in Sλ⊗SμS_\lambda\otimes S_\mu.

The paper notes that a combinatorial description has been sought since 1938 and that deciding whether a Kronecker coefficient is nonzero is NP-hard. The machine-learning experiments address the binary classification problem of predicting whether a coefficient vanishes, but do not provide the requested general combinatorial description.

References

In the context of combinatorial representation theory, one can ask: Can we give a combinatorial description of the Kronecker coefficients? This question has remained open since 1938.

— Mathematical Data Science  (2502.08620 - Douglas et al., 12 Feb 2025) in Section 3, subsection “Kronecker coefficients”

In stark contrast to the analogous Littlewood--Richardson coefficients for $GL_N(\mathbb C)$, no combinatorial description of $g_{\lambda, \mu}\nu$ has been found since Murnaghan first posed the question in 1938. This remains one of the central open problems in combinatorial representation theory, with only partial results available (e.g., ).

— Interpretable Machine Learning for Kronecker Coefficients  (2502.11774 - Butbaia et al., 17 Feb 2025) in Section 2, subsection “Kronecker coefficients” (Preliminaries)