In two papers, Burgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper we show that deciding positivity of plethysm coefficients is NP-hard, and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

中文翻译：

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体积系数的计算复杂性

在两篇论文中，Burgisser 和 Ikenmeyer (STOC 2011, STOC 2013) 使用了 Mulmuley 和 Sohoni (Siam J Comput 2001, 2008) 的几何复杂性理论 (GCT) 方法的改编版本来证明矩阵乘法边界秩的下界张量。一个关键因素是有关某些 Kronecker 系数的信息。虽然张量是 GCT 思想的有趣测试平台，但远期目标是代数复杂性类的分离。Kronecker 系数在该设置中的作用由所谓的 plethysm 系数承担：这些是多项式空间的坐标环中的多重性。尽管已知 Kronecker 系数的几个硬度结果，但几乎没有关于计算体积系数甚至确定其正性的复杂性的结果。在本文中，我们表明确定体积系数的正性是 NP-hard，而计算体积系数是 #P-hard。事实上，即使 plethysm 系数的内部参数是固定的，这两个问题仍然很难解决。通过这种方式，我们获得了内部与外部的对比：如果体积系数的外部参数是固定的，那么体积系数可以在多项式时间内计算出来。此外，我们推导出新的下限和上限，在特殊情况下甚至推导出我们认为具有独立意义的体积系数的组合描述。我们的技术以比 Ikenmeyer、Mulmuley 和 Walter（Comput Compl 2017）最近关于 Kronecker 系数的工作更精细的方式使用离散断层扫描。这使我们的工作成为第一个将离散断层扫描技术应用于体积系数研究的工作。相当令人惊讶的是，这种解释还导致某些体积系数和 Kronecker 系数之间出现新的等式。