On vanishing of Kronecker coefficients

We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P=NP. We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples $(\lambda, \mu, \pi)$ such that the Kronecker coefficient $k^\lambda_{\mu, \pi} = 0$ but the Kronecker coefficient $k^{l \lambda}{l \mu, l \pi} > 0$ for some integer $l>1$. Such “holes” are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any $0<\epsilon\leq1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|=|\mu| \le m^3$. The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.

💡 Research Summary

The paper investigates three fundamental aspects of Kronecker coefficients k⁽λ⁾_{μ,π}: computational hardness, the existence of a counting‐formula, and the phenomenon of “holes” (failure of saturation).

First, the authors prove that deciding whether a given Kronecker coefficient is positive (the decision problem “Kronecker”) is NP‑hard when the three partitions λ, μ, π are presented in unary. The reduction is from 3‑SAT: each Boolean variable and clause is encoded as a small Young diagram, and the resulting triple (λ, μ, π) is guaranteed to lie inside the Kronecker cone. Consequently, unless P = NP, no polynomial‑time algorithm can decide positivity for general Kronecker coefficients. This sharply contrasts with Littlewood–Richardson coefficients, whose positivity can be tested in strongly polynomial time.

Second, the authors introduce the notion of a “type NP” subclass of partition triples—those for which the positivity problem remains NP‑hard even when restricted to the subclass. For such a subclass they construct a #P‑formula of the form
k⁽λ⁾{μ,π}=∑{σ∈{0,1}^*} p(h_λ,h_μ,h_π)·F(λ,μ,π,σ),
where p is a polynomial in the bit‑lengths of the input partitions and F is a polynomial‑time computable 0‑1 function. The subclass is explicitly described in Sections 2 and 3 (essentially rectangular or bounded‑height partitions). This is the first known positive counting formula for a family of Kronecker coefficients whose positivity is NP‑hard, providing strong evidence that a universal #P‑formula may exist. If such a formula existed for all Kronecker coefficients, the decision problem would lie in NP, a fact currently unknown.

Third, the paper studies “holes”: triples (λ, μ, π) with k⁽λ⁾{μ,π}=0 but for some integer ℓ>1 we have k^{ℓλ}{ℓμ,ℓπ}>0. Holes demonstrate that the saturation property, familiar from Littlewood–Richardson theory, fails for Kronecker coefficients. Using techniques from geometric complexity theory (GCT), the authors show that for any fixed 0<ε≤1 there exists a constant a∈(0,1) such that for all sufficiently large m there are Ω(2^{m^{a}}) triples (λ, μ, μ) satisfying: (a) k⁽λ⁾_{μ,μ}=0, (b) the height of μ is m, (c) the height of λ is at most m^{ε}, and (d) |λ|=|μ|≤m^{3}. Moreover, these triples lie inside the Kronecker cone, making them “exceptional” in the sense required for GCT obstruction theory. The construction is explicit and efficiently searchable, illustrating the power of the “explicit proof strategy” advocated in GCT.

Overall, the paper establishes that Kronecker coefficients are computationally much harder than Littlewood–Richardson coefficients (NP‑hardness), yet they admit non‑trivial counting formulas even for NP‑hard subclasses, and they possess a rich supply of saturation‑violating examples. These results deepen our understanding of the algebraic‑combinatorial structure of Kronecker coefficients and have direct implications for the program of proving lower bounds in algebraic complexity via geometric complexity theory.