Coxeter Groups and Unimodality — Problem 10

Definitions and mathematical statement

The supplied record does not define $\tau(G;x)$ or specify the class of graphs under consideration. Let $\mathcal G$ denote this unspecified intended graph class. Once $\tau$ and $\mathcal G$ have been defined, the question is whether \[ \forall G\in\mathcal G\;\forall z\in{\mathbb{C}},\qquad \tau(G;z)=0\ \Longrightarrow\ z\in{\mathbb{R}}. \] Thus the intended assertion is that $\tau(G;x)$ has only real roots for every admissible graph $G$. This is a conditional transcription of the question, not a complete formulation: the definition of $\tau$, its conventions, and the admissible graphs must be recovered from the cited source before mathematical review.

Short English statement

Does the expression denoted by $\tau(G;x)$ have only real roots for every graph in the intended class? The supplied record leaves both $\tau$ and that class undefined.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800372$, exact source code catalogue entry, title "Coxeter Groups and Unimodality --- Problem 10." The registry attributes the problem to Brenti, with proposed year 2024, and labels it open; this label is supplied metadata, not an independently verified status assessment.
• Brenti, Some Open Problems on Coxeter Groups and Unimodality (2024), Problem 10. Supplied source URL: https://arxiv.org/abs/2410.09897. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access.
• The supplied statement says that the problem was raised in reference [BreChro, Prob. 7.1]. No bibliographic identification of [BreChro] is supplied.