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

2016/03/15 by Tadashi Ishibe, Ishibe, Tadashi · 2 citations
Computer Science · Mathematics · #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras #math.CO #math.RT #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1603.04517

11pages

arxiv created 2016/03/15 · arxiv updated 2016/03/16

Abstract

Let G_M⁺ be the Artin monoid of finite type generated by the letters a_i, i∈ I with respect to a Coxeter matrix M that is equipped with the degree map ° : G_M⁺ → \Z_≥0 defined by assigning to each equivalence class of words the length of the words, and let N_{M, °}(t) := ∑_{J ⊂ 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.

Citations

• Artin-Gruppen und Coxeter-Gruppen

Cited by

• The Zero Locus of the F-triangle
• Zero loci of skew-growth functions for dual Artin monoids

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