Kronecker Coefficients: #P-Hard Generally, Polynomial for Hook-Like Partitions — E8 Intelligence Research

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Kronecker coefficients are #P-hard to compute in general but can be evaluated in polynomial time when one partition is hook-like, with explicit complexity bounds. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22294712

September 4, 2026Open Access

Kronecker Coefficients: #P-Hard Generally, Polynomial for Hook-Like Partitions — E8 Intelligence Research

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Key Points

Determine the computational complexity of calculating Kronecker coefficients and establish parameter bounds under which polynomial-time algorithms exist.
Analyzed tensor product multiplicities of irreducible representations in the symmetric group $S_n$.
Derived algorithmic complexity bounds based on partition length $\ell$, largest part $N$, and second-smallest part $M$.
Demonstrated that general Kronecker coefficient computation is #P-hard and classified within $\text{FP}^{\#\text{P}}$.
Proved that hook-like partitions where $M = O(1)$ allow polynomial-time computation bounded by $O(\ell^3 N^3)$, with fixed-length partitions scaling at $N^{O(\ell)}$.

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Implication

Mathematical analysis reveals polynomial-time computability for hook-like Kronecker coefficients despite general #P-hardness, indicating tractable boundaries in representation theory.

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Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9d1ee128139818eab22007 https://doi.org/10.5281/zenodo.22294712

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