Let (W,S) be a Coxeter system with positive roots Phi^+. For biclosed subsets A,B⊆Phi^+ whose join exists in the biclosed-set order, Dyer's conjecture asserts that the join is the expected closure of their union, namely A vee B=(A cup B)^(2), where (·)^(2) denotes 2-closure. Equivalently, A vee B is described by the increasing Bruhat paths whose reflection labels lie in A cup B. The conjecture is posed for arbitrary Coxeter systems; the cited 2026 preprint claims a proof in all finite Coxeter groups.

Dyer’s conjecture on joins of biclosed sets

Let $(W,S)$ be a Coxeter system with positive roots $\Phi^+$ . For biclosed subsets $A,B\subseteq\Phi^+$ whose join exists in the biclosed-set order, Dyer's conjecture asserts that the join is the expected closure of their union, namely $A\vee B=(A\cup B)^{(2)}$ , where $(\cdot)^{(2)}$ denotes $2$ -closure. Equivalently, $A\vee B$ is described by the increasing Bruhat paths whose reflection labels lie in $A\cup B$ . The conjecture is posed for arbitrary Coxeter systems; the cited 2026 preprint claims a proof in all finite Coxeter groups.

References

Primary source

arXiv

Additional references

Bounded joins of biclosed sets — arXiv — Yibo Gao, Yulin Peng, Hanlin Xu

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture for all finite symmetry types, but the claim has not been independently verified and the broader infinite case remains open.

Dyer’s conjecture concerns whether joins of biclosed root sets have the expected closure description. The tracked result is restricted to finite Coxeter groups; the original conjecture for arbitrary Coxeter groups is broader.

Known results

Dyer (2011) established the weak-order and biclosed-set framework, including the join description when joins exist.
An October 2025 paper proved the conjecture in types $A$ and $I$ , and reported computational checks in types $H_{3}$ and $F_{4}$ .
A September 2026 paper reported proofs for all simply-laced finite types, including $A$ , $D$ , and $E$ .

September 2026 claimed uniform proof

Yibo Gao, Yulin Peng, and Hanlin Xu claim a type-uniform proof for all finite Coxeter groups, together with a stronger closure theorem for $2$ -coclosed sets. This would settle the finite case, but the claim is unverified; another September source describes the all-finite-type uniform approach as unresolved.

Current status (as of September 2026): The conjecture is claimed proved for all finite Coxeter groups, but that claim is unverified; the formulation for arbitrary Coxeter groups remains open.

Sources

No solutions have been posted yet.