On the derivative at $t = 1$ of the skew-growth functions for Artin monoids

Abstract

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 $\deg\!:\!G_{M}^{+} \to\!\Z_{\ge0}$ defined by assigning to each equivalence class of words the length of the words, and let $N_{M, \deg}(t)\!:=\!\!\sum_{J \subset I}(-1)^{\#J} t^{\deg(\Delta_{J})}$ be the skew-growth function, where the summation index $J$ runs over all subsets of $I$ and $\Delta_{J}$ 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 $N_{M, \deg}(t)$. As a result, we show that the polynomial $N_{M, \deg}(t)$ has a simple root at $t = 1$.