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On vanishing of Kronecker coefficients
computational complexity ( IF 0.7 ) Pub Date : 2017-07-27 , DOI: 10.1007/s00037-017-0158-y
Christian Ikenmeyer , Ketan D. Mulmuley , Michael Walter

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 $${(\lambda, \mu, \pi)}$$(λ,μ,π) such that the Kronecker coefficient $${k^\lambda_{\mu, \pi} = 0}$$kμ,πλ=0 but the Kronecker coefficient $${k^{l\lambda}_{l \mu, l \pi} > 0}$$klμ,lπlλ>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 $${0 < \epsilon \leq 1}$$0<ϵ≤1, there exists $${0 < a < 1}$$0

中文翻译:

关于 Kronecker 系数的消失

我们表明,决定 Kronecker 系数的正性的问题是 NP-hard 问题。以前,这个问题被推测在 P 中,就像 Littlewood-Richardson 系数一样。我们的结果以形式化的方式确立了 Kronecker 系数比 Littlewood-Richardson 系数更难,除非 P = NP。我们还表明存在一个 #P 公式,用于 Kronecker 系数的特定子类,其正性是 NP 难以确定的. 这是一个证据,尽管正问题的难度很大,但很可能存在克罗内克系数的正组合公式。找到这样的公式是表示论和代数组合学中的一个主要开放问题。 最后,我们考虑划分三元组 $${(\lambda, \mu, \pi)}$$(λ,μ, π) 使得 Kronecker 系数 $${k^\lambda_{\mu, \pi} = 0}$$kμ,πλ=0 但 Kronecker 系数 $${k^{l\lambda}_{l \mu , l \pi} > 0}$$klμ,lπlλ>0 对于一些整数 l > 1。这样的“洞”很有趣,因为它们见证了克罗内克系数饱和特性的失败,这仍然知之甚少。使用计算复杂性理论的见解,我们将我们的硬度证明转化为积极的结果:我们表明不仅存在许多这样的三元组,而且还可以有效地找到它们。具体来说,我们证明,对于任何 $${0 < \epsilon \leq 1}$$0<ϵ≤1,存在 $${0 < a < 1}$$0 这种“洞”引起了人们极大的兴趣,因为它们见证了克罗内克系数饱和特性的失效,而这一点仍然知之甚少。使用计算复杂性理论的见解,我们将我们的硬度证明转化为积极的结果:我们表明不仅存在许多这样的三元组,而且还可以有效地找到它们。具体来说,我们证明,对于任何 $${0 < \epsilon \leq 1}$$0<ϵ≤1,存在 $${0 < a < 1}$$0 这种“洞”引起了人们极大的兴趣,因为它们见证了克罗内克系数饱和特性的失效,而这一点仍然知之甚少。使用计算复杂性理论的见解,我们将我们的硬度证明转化为积极的结果:我们表明不仅存在许多这样的三元组,而且还可以有效地找到它们。具体来说,我们证明,对于任何 $${0 < \epsilon \leq 1}$$0<ϵ≤1,存在 $${0 < a < 1}$$0
更新日期:2017-07-27
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