Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(\alpha,\beta,\gamma)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(N\alpha,N\beta,N\gamma)>0$ implies $\overline{g}(\alpha,\beta,\gamma)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(\alpha,\beta,\gamma)$, over all $|\alpha|+|\beta|+|\gamma| = n$. Finally, we show that computing $\overline{g}(\lambda,\mu,\nu)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(\lambda,\mu,\nu)$ is in unary.


Introduction
The reduced Kronecker coefficients were introduced by Murnaghan in 1938 as the stable limit of Kronecker coefficients, when a long first row is added: see [Lit58].As such, they occupy the middle ground between the Kronecker and the LR-coefficients.While the latter are well understood and have a number of combinatorial interpretations, the former are notorious for their difficulty (cf.[Kir04, Prob.2.32]).It is generally believed that the reduced Kronecker coefficients are simpler and more accessible than the (usual) Kronecker coefficients, cf.[Kir04,OZ19].The results of this paper suggest otherwise, see §4.1.
Theorem 2. For all k ≥ 3, the triple of partitions 1 k 2 −1 , 1 k 2 −1 , k k−1 is a counterexample to Conjecture 1.Moreover, for every partition γ s.t.γ 2 ≥ 3, there are infinitely many pairs (a, b) ∈ N 2 for which the triple of partitions (a b , a b , γ) is a counterexample to Conjecture 1.
These results both contrast and complement [CR15, Cor.6], which confirms the saturation property for triples of the form (a b , a b , a).

Maximal values.
Our second result is a variation on Stanley's recent bound on the maximal Kronecker and LR-coefficients: Sta16,Sta17], see also [PPY19]).We have: In [PPY19], we prove that the maximal Kronecker and LR-coefficients appear when all three partitions have near-maximal dimension, which in turn implies that they have a Vershik-Kerov-Logan-Shepp (VKSL) shape.See also [PP20+] for refined upper bounds on (reduced) Kronecker coefficients with few rows.Here we obtain the following analogue of Stanley's Theorem 3.
Here by strongly #P-hard we mean #P-hard when the input (λ, µ, ν) is given in unary.In other words, the input size of the problem is the total number of squares in the three Young diagrams.The theorem is in sharp contrast with computing χ (n−k,k) [λ] which is #P-complete but not strongly #P-complete, see [PP17, §7].Theorem 6. Computing the reduced Kronecker coefficients g(α, β, γ) is strongly #P-hard.
Let us mention that the problem of computing the (reduced) Kronecker coefficients is not known to be in #P, see [PP17].In fact, finding a combinatorial interpretation for (reduced) Kronecker coefficients is a classical open problem [Sta00, Prob.10].Note also that Theorem 6 is stronger than Theorem 5, since in the limit (•) it suffices to take n ≥ |α| + |β| + |γ|, see [BOR11,Val99].Indeed, this shows that the reduced Kronecker coefficient problem is a subset of instances of the usual Kronecker coefficient problem (cf., however, §4.6).

Disproof of the saturation property
2.1.Preliminaries.We assume the reader is familiar with basic results and standard notations in Algebraic Combinatorics, see [Sag01,Sta99].We also need the following two results on Kronecker coefficients.
Let ℓ(λ) be the number of parts of the partition λ, and d(λ) := max{k : λ k ≥ k} be the Durfee size.
The following argument gives a blueprint for the proof of Theorem 2.
On the other hand, let N ≥ 3ℓ 2 /a, ν := N γ, r := b + 1, and s := N a.Then |ν| ≤ N ab/6 < rs/6, r > 3ℓ 3/2 , and s = N a ≥ 3ℓ 2 , by construction.Since ν ∈ X for all N > 1, the conditions of Lemma 12 are satisfied.We conclude: which implies that (α, α, γ) is a counterexample to the saturation property.Since the construction works for all b large enough as above, this proves the second part of the theorem.For the upper bound, by [PPY19, Thm 1.5] which extends ( * * ) in Theorem 3, we have:

Bounds and complexity via identities
Using the Vandermonde identity for the sums of binomial coefficients, we have: In this notation, the theorem is a maximum over a + b + q ≤ 3n.Combining these with ( * ) in Theorem 3, we have: n) .For the lower bound, let α, β, γ ⊢ n, and note that g(α, β, γ) ≥ g(α, β, γ), which is achieved in (3.1) for m = 0.The result now follows from part ( * ) of Theorem 3.

Final remarks and open problems
4.1.All three results in this paper are centered around the same (philosophical) claim, that the reduced Kronecker coefficients are closer in nature to the (usual) Kronecker coefficients than to the LR-coefficients.This is manifestly evident from both the statements and the proofs of the theorems.In fact, the only clue we know of the difference is the result in [Ent16].However, this claim should not be taken as a suggestion that the LR-coefficients are not strongly #P-hard.We do, in fact, conjecture that computing c λ µν is strongly #P-hard [PP17, Conj.8.1], but this remains beyond the reach of existing technology.4.2.There is a general setting which extends the stability of Kronecker coefficients to other families of stable limits, see [SS16].Manivel asks if the saturation property holds for all these families, but notes that "we actually have only very limited evidence for that" [Man15].In view of our results, it would be interesting to see if the saturation property holds for any of these families of stable coefficients.4.3.Theorem 5 is not stated in [IMW17] in this form.It does however follow directly from the proof, which is essentially a parsimonious reduction from the 3-Partition problem classically known to be (strongly) NP-complete, and thus the counting is (strongly) #P-complete.

By analogy with [PPY19]
, it would be interesting to find the asymptotic limit shape of partitions α, β, γ which achieve a maximum in Theorem 4. We believe the bounds in our proof of the theorem can be improved to show that all three shapes are Plancherel of the same size.4.5.Among other consequences, the saturation property implies that the vanishing problem c λ µν > ?0 is in P, see [MNS12].The main result of [IMW17] proved that the vanishing problem g(λ, µ, ν) > ?0 is NP-hard, refuting Mulmuley's conjecture (see e.g.[PP17,§2]).Following the pattern in §4.1 above, we conjecture that the vanishing problem g(α, β, γ) > ?0 for reduced Kronecker coefficients is also NP-hard.4.6.There is a subtle but important technical differences between the way we state Theorem 5 and the way it is stated in [IMW17].While we implicitly use the (standard) Turing reduction to derive Theorem 6 from Theorem 5, the original proof in [IMW17] uses a more restrictive many-to-one reduction (sometimes called Karp reduction).Such a reduction for Theorem 6 would also resolve our conjecture above on the vanishing problem for the reduced Kronecker coefficients.4.7.In an appendix to [BOR09], Mulmuley conjectures a weaker property than the saturation, for the stretched Kronecker coefficients, which would in turn imply a polynomial result for a closely related complexity problem.See also [Kir04] for further saturation related conjectures.