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根系排屑Ⅰ：间歇性排屑

Root system chip-firing I: Interval-firing

Pavel Galashin, Sam Hopkins, Thomas McConville, Alexander Postnikov

备注：
54 pages, 12 figures, 2 tables; v2: major revisions to improve exposition; v3: to appear in Mathematische Zeitschrift (Math. Z.)
发表信息：
Mathematische Zeitschrift, 292(3-4), 2019
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摘要：

Jim Propp最近在一条线上引入了一种芯片触发的变体，其中芯片被赋予不同的整数标签。Hopkins、McConville和Propp表明，这个过程是从一些（但不是所有）芯片的初始配置融合而来的。我们根据根系统重新构建了它们的设置：标记的芯片发射可以被看作是一个根发射过程，该过程允许$α\in Φ^{+}$的移动$λ\to λ+ α$，只要$\langleλ,α^\vee\rangle = 0$，其中$Φ^{+}$是A型根系的正根的集合，$λ$是该根系的权重。因此，我们的动机是研究完全相同的根放电过程中的任意根系。实际上，这个中央神经根放电过程是本文后续部分的主题。在本文中，我们反而研究了$λ\to λ+ α$为$α\in Φ^{+}$确定的间隔根放电过程，无论何时$\langleλ,α^\vee\rangle \in [-k-1,k-1]$或$\langleλ,α^\vee\rangle \in [-k,k-1]$，对于任何$k \geq 0$。我们证明，这些间隔射击过程总是合流，从任何初始重量。我们还表明，有一个自然的方式来始终标记这些间隔射击过程的稳定点在所有值的$k$，使给定的稳定的权重的数量是一个多项式在$k$。我们猜想，这些Ehrhart-like多项式有非负整数系数。

英文摘要：

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $λ\to λ+ α$ for $α\in Φ^{+}$ whenever $\langleλ,α^\vee\rangle = 0$, where $Φ^{+}$ is the set of positive roots of a root system of Type A and $λ$ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by $λ\to λ+ α$ for $α\in Φ^{+}$ whenever $\langleλ,α^\vee\rangle \in [-k-1,k-1]$ or $\langleλ,α^\vee\rangle \in [-k,k-1]$, for any $k \geq 0$. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of $k$ so that the number of weights with given stabilization is a polynomial in $k$. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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