Coxeter Groups and Unimodality — Conjecture 5

Definitions and mathematical statement

Let $(W,S)$ be a Coxeter system. The supplied record does not define the relation $\leq$. Under the conventional interpretation as strong Bruhat order, $u\leq v$ means that a reduced expression for $v$ contains a subword that is a reduced expression for $u$.

Let $c,d$ be noncommuting indeterminates, and let $\{c,d\}^{*}$ denote the set of finite words in them, including the empty word. Define \[ \mathbb{Z}_{\geq 0}\langle c,d\rangle = \left\{\sum_{w\in\{c,d\}^{*}} a_w w: a_w\in\mathbb{Z}_{\geq 0},\quad a_w=0\text{ for all but finitely many }w\right\}, \] with multiplication induced by concatenation of words. This makes explicit the nonnegative-coefficient interpretation of the source notation $\mathbb{N}\langle c,d\rangle$.

Conjecture 5 (Brenti, 2024; conditional transcription). For every Coxeter system $(W,S)$ and every $u,v\in W$ with $u\leq v$, \[ \widetilde{\Phi}_{u,v}\in\mathbb{Z}_{\geq 0}\langle c,d\rangle. \] Thus, if the source invariant is written as a noncommutative polynomial \[ \widetilde{\Phi}_{u,v}=\sum_{w\in\{c,d\}^{*}} a_w(u,v)w, \] the assertion is that every coefficient $a_w(u,v)$ is a nonnegative integer.

The supplied record does not define $\widetilde{\Phi}_{u,v}$, its construction, or its normalization. Consequently, it does not by itself determine the invariant whose coefficients are asserted to be nonnegative. The Bruhat-order interpretation above also requires confirmation from the cited source. No finiteness restriction on $W$ or $S$ is stated. The quantifier includes $u=v$, but the value or convention for $\widetilde{\Phi}_{u,u}$ is not supplied.

Short English statement

Brenti's supplied Conjecture 5 asserts that, for every comparable pair in every Coxeter system, $\widetilde{\Phi}_{u,v}$ is a polynomial in noncommuting $c,d$ with nonnegative integer coefficients. The invariant is undefined in the supplied record, and interpreting comparability as strong Bruhat order requires confirmation. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800367$, source problem code catalogue entry, title "Coxeter Groups and Unimodality --- Conjecture 5," in the AMR Open Problem Lists collection. The record attributes the conjecture to Brenti and gives the proposed year as 2024.
• Brenti, Some Open Problems on Coxeter Groups and Unimodality (2024), Conjecture 5. 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 statements, not independently verified access or extraction.
• The local record labels the problem "open" and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry curation, not an independently verified literature assessment.