Brenti’s log-concavity conjecture for \widetilde{R}-polynomials

Conjecture 2 (Brenti). Let $W$ be a finite Coxeter group. For $u,v\in W$ with $u\leq v$ in Bruhat order, define $\widetilde{R}_{u,v}$ by

q^{\ell(u,v)/2}\widetilde{R}_{u,v}(q^{1/2}-q^{-1/2})=R_{u,v}(q),

where $\ell(u,v)=\ell(v)-\ell(u)$ . Since all powers of $q$ in $\widetilde{R}_{u,v}(q)$ have the same parity as $\ell(u,v)$ , write

\widetilde{R}_{u,v}(q)=\begin{cases}Q_{u,v}(q^{2})&\text{if $\ell(u,v)$ is even},\\ qQ_{u,v}(q^{2})&\text{if $\ell(u,v)$ is odd},\end{cases}

with $Q_{u,v}(q)=\sum_i a_iq^i$ . Then $Q_{u,v}$ is log-concave; that is, $a_i^2\geq a_{i-1}a_{i+1}$ for every applicable index $i$ .

Progress summary

Solved

A new preprint claims the longstanding conjecture is false by giving an explicit counterexample, but the claim has not yet been independently verified.

Brenti’s conjecture concerns log-concavity of coefficient sequences of $\widetilde{R}$ -polynomials. Christian Gaetz claims to disprove it with an explicit counterexample in an August 2026 preprint.

August 2026 counterexample

Christian Gaetz’s version-one preprint claims that the conjecture is false. The counterexample and its verification remain unrefereed and subject to scrutiny.

Current status (as of August 2026): A counterexample is claimed in a new preprint, but the conjecture is not yet regarded as definitively settled because the claim remains unverified.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A counterexample to a log-concavity conjecture of Brenti — arXiv — Christian Gaetz

algebraic-combinatorics polynomial-theory

4 days ago