The Real-Rootedness and Log-concavities of Coordinator Polynomials of Weyl Group Lattices

TL;DR

This paper proves the real-rootedness of coordinator polynomials for Weyl group lattices of types A, C, and D using a new trigonometric approach, and shows that type B polynomials are generally not real-rooted, concluding all are log-concave.

Contribution

It introduces a trigonometric substitution method to establish real-rootedness for type D coordinator polynomials and analyzes the root properties across all Weyl group lattice types.

Findings

Type A and C coordinator polynomials are real-rooted.

Type D coordinator polynomials are real-rooted via new method.

Type B coordinator polynomials are not generally real-rooted.

Abstract

It is well-known that the coordinator polynomials of the classical root lattice of type $A_n$ and those of type $C_n$ are real-rooted. They can be obtained, either by the Aissen-Schoenberg-Whitney theorem, or from their recurrence relations. In this paper, we develop a trigonometric substitution approach which can be used to establish the real-rootedness of coordinator polynomials of type $D_n$. We also find the coordinator polynomials of type $B_n$ are not real-rooted in general. As a conclusion, we obtain that all coordinator polynomials of Weyl group lattices are log-concave.