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 $°:G_M^{+}\to {\mathbb{Z}}_{\ge 0}$ defined by assigning to each equivalence class of words the length of the words, and let $N_{M,°}(t):={\sum }_{J\subset I}(-1{)}^{\#J}{t}^{°({\Delta }_J)}$ 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 $N_{M,°}(t)$. As a result, we show that the polynomial $N_{M,°}(t)$ has a simple root at $t=1$.

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