On vanishing of Kronecker coefficients

Abstract

Abstract. 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 (λ, µ, π) such that the Kronecker coefficient k λ µ,π = 0 but the Kronecker coefficient k lλ lµ,lπ > 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 < ≤ 1, there exists 0 < a < 1 such that, for all m, there exist Ω(2 m a ) partition triples (λ, µ, µ) in the Kronecker cone such that: (a) the Kronecker coefficient k λ µ,µ is zero, (b) the height of µ is m, (c) the height of λ is ≤ m , and (d) |λ| = |µ| ≤ m 3 . The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.

Efficient Algorithms for Tensor Scaling, Quantum Marginals, and Moment Polytopes

“…so the solution to the one-body quantum marginal problem captures precisely the asymptotic support of the Kronecker coefficients [CM06, Kly06,CHM07]. We note that the problem of deciding whether gpλ, µ, νq ą 0 is known to be NP-hard [IMW17]. However since the asymptotic vanishing of Kronecker coefficients is captured by the quantum marginal problem, it has been conjectured that it should have a polynomial time algorithm and we make progress towards this question.…”

“…P pXq ‰ 0. In fact, it is known that even deciding the existence of highest-weight vectors is NP-hard [IMW17]. Our algorithm does not solve the membership problem via the dual description provided by Mumford's theorem.…”

Tensor models, Kronecker coefficients and permutation centralizer algebras

“…The actual evaluation of the LR coefficient for general Young diagrams is # P-hard [80] (# P is the analog of NP when we go from decision problems to counting problems). Recently it was found that deciding the vanishing of Kronecker coefficients is NP-hard [81]. This is an interesting contrast between central correlators in the 1-matrix problem (224) and the one-point function of central observables in the tensor model (79).…”

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

“…For partitions λ, µ, ν of nd let g(λ, µ, ν) ∈ N denote the Kronecker coefficient, i.e., the multiplicity of the irreducible S nd -representation [λ] in the tensor product [µ] ⊗ [ν], where [µ] ⊗ [ν] is interpreted as an S ndrepresentation via the diagonal embedding S nd ֒→ S nd × S nd , π → (π, π). A combinatorial interpretation of g(λ, µ, ν) is known only in special cases, see [Las80,Rem89,Rem92,RW94,Ros01,BO07,Bla12,Liu14,IMW15,Hay15], and finding a general combinatorial interpretation is problem 10 in Stanley's list of positivity problems and conjectures in algebraic combinatorics [Sta00]. In geometric complexity theory the main interest is focused on rectangular Kronecker coefficients, i.e., the coefficients g(λ, n × d, n × d).…”