Zero loci of skew-growth functions for dual Artin monoids

TL;DR

This paper investigates the zeros of skew-growth functions in dual Artin monoids, revealing their distribution and connection to orthogonal polynomials, with implications for understanding their asymptotic behavior.

Contribution

It establishes the exact number of real zeros of skew-growth functions for dual Artin monoids and links these functions to Jacobi polynomials, providing new insights into their structure.

Findings

Exactly rank(P) simple real zeros in (0,1] for types A_l, B_l, D_l

Skew-growth functions relate to Jacobi polynomials for types A_l and B_l

Smallest root approaches zero as rank l increases

Abstract

We show that the skew-growth function of a dual Artin monoid of finite type P has exactly rank(P) =: l simple real zeros on the interval (0, 1]. The proofs for types A_l and B_l are based on an unexpected fact that the skew-growth functions, up to a trivial factor, are expressed by Jacobi polynomials due to a Rodrigues type formula in the theory of orthogonal polynomials. The skew-growth functions for type D_l also satisfy Rodrigues type formulae, but the relation with Jacobi polynomials is not straightforward, and the proof is intricate. We show that the smallest root converges to zero as the rank l of all the above types tend to infinity.