Brenti's conjecture on special matchings and Kazhdan–Lusztig R-polynomials

Let $W$ be a Coxeter group of type $A$ , let $u,v\in W$ , and let $M$ be a special matching of the Bruhat interval $[u,v]$ . For $x,y\in [u,v]$ with $x\le y$ , write $R_{x,y}(q)$ for the Kazhdan–Lusztig $R$ -polynomial. Brenti's conjecture. For all such $x,y$ , one has

R_{x,y}(q)=\begin{cases} R_{M(x),M(y)}(q) & \operatorname{if } M(x)\lhd x,\ M(y)\lhd y,\\ R_{M(x),M(y)}(q) & \operatorname{if } M(x)\rhd x,\ M(y)\rhd y,\\ (q-1)R_{x,M(y)}(q)+qR_{M(x),M(y)}(q) & \operatorname{if } M(x)\rhd x,\ M(y)\lhd y,\\ q^{-1}R_{M(x),M(y)}(q)+(q^{-1}-1)R_{M(x),y}(q) & \operatorname{if } M(x)\lhd x,\ M(y)\rhd y. \end{cases}

The conjecture extends Brenti's result for intervals starting from the identity to arbitrary Bruhat intervals in type $A$ . The paper's abstract says that this conjecture is proved, so its status is recorded as solved; the displayed assertion is retained as the source statement.

Brenti’s 2003 conjecture extends the known identity-starting recursion for Kazhdan–Lusztig $R$ -polynomials to arbitrary Bruhat intervals in type $A$ . It prescribes four formulas according to whether a special matching moves each endpoint up or down.

Known results

Lower Bruhat intervals were explicitly characterized and classified, with applications to parabolic Kazhdan–Lusztig polynomials (2016).
For special matchings of $[e,w]$ , recursive formulas for $R_{u,w}(q)$ were established (2017), covering the identity-starting case.

June 2026 claimed proof

The preprint Special Matchings, Brenti's Conjecture, and the Combinatorial Invariance Conjecture claims a complete classification of special matchings on arbitrary type- $A$ Bruhat intervals and derives Brenti’s conjecture for all $x,yin [u,v]$ with $x\le y$ . It does not claim to prove the broader Combinatorial Invariance Conjecture, which it presents as still open.

Current status (as of September 2026): Brenti’s conjecture is claimed proved by the June 2026 preprint, but that proof remains unverified; the broader Combinatorial Invariance Conjecture remains open.

Brenti's conjecture on special matchings and Kazhdan–Lusztig $R$ -polynomials

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Progress summary

Solutions 0