Artin幺半群的斜增长函数在$t = 1$的导数
On the derivative at $t = 1$ of the skew-growth functions for Artin monoids
摘要：
令$G_{M}^{+}$是由字母$a_i, i\in I$相对于Coxeter矩阵$M$生成的有限类型的Artin幺半群，该Coxeter矩阵$M$配备有通过向每个等价类的字分配字的长度而定义的度映射$°\!:\!G_{M}^{+} \to\!\Z_{\ge0}$，并且令$N_{M, °}(t)\!:=\!\!\sum_{J \subset I}(-1)^{\#J} t^{°(Δ_{J})}$是偏斜增长函数，其中求和索引$J$在$I$的所有子集上运行，并且$Δ_{J}$是与集合相关联的$G_{M}^{+}$中的基本元素$J$。在本文中，我们将计算多项式$N_{M, °}(t)$在$t = 1$处的导数。结果表明，多项式$N_{M, °}(t)$在$t = 1$有一个单根.
英文摘要：
Let $G_{M}^{+}$ be the Artin monoid of finite type generated by the letters $a_i, i\in I$ with respect to a Coxeter matrix $M$ that is equipped with the degree map $°\!:\!G_{M}^{+} \to\!\Z_{\ge0}$ defined by assigning to each equivalence class of words the length of the words, and let $N_{M, °}(t)\!:=\!\!\sum_{J \subset I}(-1)^{\#J} t^{°(Δ_{J})}$ be the skew-growth function, where the summation index $J$ runs over all subsets of $I$ and $Δ_{J}$ is the fundamental element in $G_{M}^{+}$ associated to the set $J$. In this article, we will calculate the derivative at $t = 1$ of the polynomial $N_{M, °}(t)$. As a result, we show that the polynomial $N_{M, °}(t)$ has a simple root at $t = 1$.
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