A positive formula for the Ehrhart-like polynomials from root system chip-firing
A positive formula for the Ehrhart-like polynomials from root system chip-firing
复制标题
根系芯片烧成的类埃尔哈特多项式的正公式
作者：
S. Hopkins;A. Postnikov
在与 Pavel Galashin 和 Thomas McConville 的早期合作中，我们引入了根系统的芯片激发版本。我们对根系芯片烧制的研究使我们定义了某些类似于晶格多面体埃尔哈特多项式的多项式，我们将其称为对称且截断的类埃尔哈特多项式。我们推测这些多项式具有非负整数系数。在这里，我们通过提供对称类埃尔哈特多项式的系数的正组合公式来肯定这个正性猜想的“一半”。该公式取决于全六面体切片的微妙完整性属性，以及有关根多面体投影膨胀的引理，两者都可能具有独立的意义。我们还讨论了我们的公式如何非常自然地提出对截断类埃尔哈特多项式的系数的猜想，该猜想通常被证明是错误的，但在某些情况下可能成立。
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart polynomials of lattice polytopes, which we termed the symmetric and truncated Ehrhart-like polynomials. We conjectured that these polynomials have nonnegative integer coefficients. Here we affirm "half" of this positivity conjecture by providing a positive, combinatorial formula for the coefficients of the symmetric Ehrhart-like polynomials. This formula depends on a subtle integrality property of slices of permutohedra, and in turn a lemma concerning dilations of projections of root polytopes, which both may be of independent interest. We also discuss how our formula very naturally suggests a conjecture for the coefficients of the truncated Ehrhart-like polynomials that turns out to be false in general, but which may hold in some cases.
根系统芯片点火 II：中央点火
影响因子：
1
作者：
Galashin, Pavel;Hopkins, Sam;McConville, Thomas;Postnikov, Alexander
通讯作者：
Postnikov, Alexander