Coxeter Groups and Unimodality — Conjecture 4

Definitions and mathematical statement

Let $W$ be a Weyl group of type $A$, equipped with its standard Coxeter generators and Bruhat order $\leq$. For $u,v\in W$ with $u\leq v$, write \[ [u,v]=\{x\in W:u\leq x\leq v\}. \] Let $P_{u,v}(q)$ denote the ordinary Kazhdan--Lusztig polynomial, with $q$ an indeterminate and normalization $P_{w,w}(q)=1$.

The supplied statement of Brenti's Conjecture 4 asserts that, for every such $W,u,v$ and every $z\in[u,v]$ for which $[u,z]$ is a hypercube decomposition of $[u,v]$, \[ P_{u,v}(q)=\widetilde{P}_{u,v,z}(q). \] The equality is an identity in the indeterminate $q$.

The supplied record does not define either the condition that $[u,z]$ be a hypercube decomposition of $[u,v]$ or the expression $\widetilde{P}_{u,v,z}(q)$. These source-specific notions remain unresolved here; in particular, the displayed assertion does not apply to every lower interval merely because it has the form $[u,z]$. A self-contained precise formulation requires the definitions and any accompanying hypotheses from the cited source.

Short English statement

Brenti's supplied conjecture says that, in every type $A$ Weyl group, the Kazhdan--Lusztig polynomial of a Bruhat interval equals the source's expression $\widetilde{P}_{u,v,z}(q)$ for every hypercube decomposition $[u,z]$ of that interval. The record omits the definitions of hypercube decomposition and of the expression on the right, so the formulation requires source clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800366$, source problem code catalogue entry, title "Coxeter Groups and Unimodality --- Conjecture 4." The supplied record attributes the conjecture to Brenti, with proposed year 2024, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Brenti, Some Open Problems on Coxeter Groups and Unimodality (2024), Conjecture 4. Supplied source URL: https://arxiv.org/abs/2410.09897. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent access or verification in preparing this draft.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry record was created and updated on 2026-10-16.