Dyer's Conjecture and Extended Weak Order

• Dyer's conjecture posits that the extended weak order, defined via biclosed subsets of positive roots, forms a complete lattice in any Coxeter group.
• It extends the classical weak Bruhat order by incorporating all biclosed sets, thereby restoring join operations lost in infinite-type groups.
• Proven in affine types and rank‑3 universal cases, the conjecture also inspires related studies on Kazhdan–Lusztig invariance and transcendental mappings.

Dyer's conjecture is a label attached to several conjectural statements associated with Matthew Dyer. In Coxeter theory, one central usage is Conjecture 1.1 that the extended weak order of any Coxeter group is a complete lattice. Closely related literature also discusses Dyer’s Conjecture D on joins in the extended weak order, Dyer’s Conjecture 2.8 on inversion sets and Bruhat preclosure, the Lusztig–Dyer combinatorial invariance conjecture for Kazhdan–Lusztig polynomials, and a transcendence conjecture attached to Dyer’s outer automorphism of $\mathrm{PGL}(2,\mathbb Z)$ (Barkley et al., 2023, Biagioli et al., 13 Oct 2025, Dermenjian, 9 Dec 2025, Burrull et al., 2021, Uludağ et al., 2016).

1. Precise Coxeter-theoretic statement

Let $(W,S)$ be a Coxeter system with root system $\Phi$, chosen positive roots $\Phi^+$, and inversion sets $\operatorname{Inv}(w)\subseteq \Phi^+$. The weak Bruhat order is characterized by inversion inclusion,

$$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$

For finite Coxeter groups, weak order is a complete lattice. For infinite Coxeter groups, it is only a meet-semilattice, and the introduction of Barkley–Speyer cites Björner’s result that weak order is never a lattice in infinite type (Barkley et al., 2023).

Dyer enlarged weak order by replacing finite inversion sets with biclosed subsets of $\Phi^+$. A subset $B\subseteq \Phi^+$ is biclosed if whenever $\alpha,\beta\in\Phi^+$ and $\gamma$ is a nonnegative linear combination of $(W,S)$0, then $(W,S)$1 and $(W,S)$2. Equivalently, the restriction of $(W,S)$3 to every rank $(W,S)$4 root subsystem is the inversion set of a region of the corresponding rank $(W,S)$5 Coxeter arrangement. The extended weak order is the poset of all biclosed subsets of $(W,S)$6, ordered by inclusion (Barkley et al., 2023).

The finite biclosed subsets of $(W,S)$7 are precisely the inversion sets of elements of $(W,S)$8. Hence weak order embeds into extended weak order as an order ideal, and in finite type the two orders coincide. Dyer’s conjecture is then:

Conjecture 1.1 (Dyer). The extended weak order of any Coxeter group is a complete lattice.

Here “complete lattice” means that every family of biclosed sets has both a join and a meet, not merely every finite family (Barkley et al., 2023).

2. Biclosedness, separability, and lattice operations

The Barkley–Speyer framework develops the conjecture for a general finite or countable set $(W,S)$9. A subset $\Phi$0 is closed if

$\Phi$1

It is coclosed if $\Phi$2 is closed, and biclosed if it is both closed and coclosed. Two stronger notions are weak separability and separability: $\Phi$3 and

$\Phi$4

for some $\Phi$5. For finite $\Phi$6, weak separability implies separability by Farkas’ lemma (Barkley et al., 2023).

This leads to the notion of a clean arrangement: $\Phi$7 is clean if every biclosed subset of $\Phi$8 is weakly separable, equivalently separable in the finite situations used in the affine proof. In clean situations, regions of the dual hyperplane arrangement

$\Phi$9

parametrize biclosed sets. The conjectural lattice problem is thereby converted into a geometric problem about arrangement regions and local separability (Barkley et al., 2023).

A suitable ordering $\Phi^+$0 is an ordering satisfying two local conditions: in every $\Phi^+$1-dimensional linear subset, the fundamental vectors come first; and for each initial segment $\Phi^+$2 and any $\Phi^+$3, there is a full subset $\Phi^+$4 containing $\Phi^+$5 such that $\Phi^+$6 is clean. If $\Phi^+$7 has a suitable ordering, then biclosed subsets of $\Phi^+$8 form a complete lattice, with explicit operations

$\Phi^+$9

A crucial reduction lemma states that any three positive roots lie in a full subsystem of rank at most $\operatorname{Inv}(w)\subseteq \Phi^+$0, so the suitability condition reduces to rank-$\operatorname{Inv}(w)\subseteq \Phi^+$1 cleanliness checks (Barkley et al., 2023).

3. Affine Coxeter groups

The principal theorem of Barkley and Speyer is:

Theorem A. The extended weak order of an affine Coxeter group is a complete lattice.

This proves Dyer’s conjecture in affine type. The paper also proves a local cleanliness theorem: if $\operatorname{Inv}(w)\subseteq \Phi^+$2 is a finite crystallographic root system or a rank $\operatorname{Inv}(w)\subseteq \Phi^+$3 untwisted affine root system, then every order ideal in the root poset on $\operatorname{Inv}(w)\subseteq \Phi^+$4 is clean. From the same machinery it deduces a unique minimal extension theorem for biclosed subsets of such order ideals, and it proves Dyer’s other affine conjecture on maximal chains: any maximal chain in the extended weak order on a finite crystallographic or untwisted affine root system is the set of initial segments of a unique total ordering on the positive roots (Barkley et al., 2023).

The proof is local and geometric rather than classificatory. Barkley–Speyer establish cleanliness directly in finite type $\operatorname{Inv}(w)\subseteq \Phi^+$5 and in rank-$\operatorname{Inv}(w)\subseteq \Phi^+$6 untwisted affine type $\operatorname{Inv}(w)\subseteq \Phi^+$7, then transfer the remaining rank-$\operatorname{Inv}(w)\subseteq \Phi^+$8 cases by folding: $\operatorname{Inv}(w)\subseteq \Phi^+$9 For finite and untwisted affine root systems, any total order refining the root poset becomes suitable, so the complete-lattice formulas above apply (Barkley et al., 2023).

The affine theorem is broader than the local cleanliness statement. The paper is careful to distinguish the proved global lattice theorem for affine Coxeter groups from stronger local order-ideal properties that fail in twisted affine settings. It also records that before this work the conjecture was known only in rank $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$0 affine groups and in affine types $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$1 and $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$2, whereas the new theorem covers all affine Coxeter groups (Barkley et al., 2023).

4. The first non-affine infinite case

A later advance proves Dyer’s conjecture for the rank $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$3 universal Coxeter group

$$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$4

This is described as the first non-spherical, non-affine Coxeter group for which Dyer’s conjecture has been proven (Barkley et al., 31 Aug 2025).

The main theorem of that paper states that every biclosed set of positive roots for $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$5 is weakly separable; as a consequence, the extended weak order for $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$6 is a lattice. A second theorem identifies arbitrary joins: for a collection $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$7 of biclosed sets,

$$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$8

where $$u\le v \Longleftrightarrow \operatorname{Inv}(u)\subseteq \operatorname{Inv}(v).$$9 is the $\Phi^+$0-closure, the smallest closed subset of $\Phi^+$1 containing $\Phi^+$2. The paper also proves an intermediate convex-closure formula,

$\Phi^+$3

and then sharpens it to $\Phi^+$4-closure (Barkley et al., 31 Aug 2025).

The proof uses features specific to $\Phi^+$5: the right Cayley graph is a trivalent tree embedded in the hyperbolic disk, each positive root corresponds to a unique edge, and faces correspond to rank $\Phi^+$6 parabolic subsystems. A biclosed partition $\Phi^+$7 induces a red–blue edge coloring. From local color-change constraints on each face, the authors construct two global curves, the “snakes”, and prove that the pair of snakes is a connected topological $\Phi^+$8-manifold separating the disk into two regions; all edges in the same region have the same color. The asymptotic behavior of the snakes yields a weakly separating hyperplane, establishing weak separability of every biclosed set (Barkley et al., 31 Aug 2025).

5. Join-description and Bruhat-preclosure conjectures

Dyer formulated further conjectures about joins. One of them, called Conjecture D in the 2025 paper of Biagioli and Perrone, predicts an explicit description of the join of biclosed sets: $\Phi^+$9 where

$B\subseteq \Phi^+$0

In finite type this is reformulated as

$B\subseteq \Phi^+$1

with $B\subseteq \Phi^+$2 the set of vertices on Bruhat paths from $B\subseteq \Phi^+$3 using only labels in $B\subseteq \Phi^+$4. The paper proves this conjecture in finite Coxeter types $B\subseteq \Phi^+$5 and $B\subseteq \Phi^+$6, and reports Sage verification in types $B\subseteq \Phi^+$7 and $B\subseteq \Phi^+$8; it explicitly distinguishes this join-description conjecture from the separate lattice conjecture (Biagioli et al., 13 Oct 2025).

Another paper studies Dyer’s Conjecture 2.8 on inversion sets and the Bruhat graph. For $B\subseteq \Phi^+$9, the Bruhat preclosure is

$\alpha,\beta\in\Phi^+$0

The conjectural formula is

$\alpha,\beta\in\Phi^+$1

The paper shows that $\alpha,\beta\in\Phi^+$2 is a preclosure, not a closure, in general; it introduces the infinite Bruhat closure

$\alpha,\beta\in\Phi^+$3

proves uniformly that

$\alpha,\beta\in\Phi^+$4

whenever the join exists, and then proves that in type $\alpha,\beta\in\Phi^+$5 the original Bruhat preclosure is already a closure, so Dyer’s original formula holds there (Dermenjian, 9 Dec 2025).

6. Other conjectures bearing Dyer’s name and the present frontier

The literature represented here attaches Dyer’s name to additional conjectures. In the affine Weyl group of type $\alpha,\beta\in\Phi^+$6, the combinatorial invariance conjecture—described as due independently to G. Lusztig and M. Dyer—asserts that isomorphic Bruhat intervals have equal Kazhdan–Lusztig polynomials,

$\alpha,\beta\in\Phi^+$7

and this is proved there. In a different direction, the paper on the involution $\alpha,\beta\in\Phi^+$8 induced by Dyer’s outer automorphism of $\alpha,\beta\in\Phi^+$9 states the conjecture that algebraic numbers of degree at least three are mapped to transcendental numbers under $\gamma$0; that paper explicitly notes that it does not formally label this statement as “Dyer’s conjecture” (Burrull et al., 2021, Uludağ et al., 2016, Barkley et al., 2023, Barkley et al., 31 Aug 2025).

Usage	Statement	Status in cited work
Extended weak order	$\gamma$1 is a complete lattice	Proved for affine Coxeter groups; proved for $\gamma$2; open in general
Kazhdan–Lusztig combinatorial invariance	$\gamma$3	Proved for $\gamma$4
Modular involution	$\gamma$5 is transcendental for algebraic $\gamma$6 of degree $\gamma$7	Conjectural

Within the extended-weak-order program, the general case remains open. Barkley–Speyer formulate the intermediate conjecture that any root system has a suitable order, and explicitly suggest this as a route toward the full conjecture. Their analysis also shows that some stronger local cleanliness and extension statements fail in twisted affine settings. The $\gamma$8 theorem suggests a second frontier: extending beyond affine and beyond planar rank-$\gamma$9 geometry to higher-rank universal Coxeter groups and, more broadly, determining whether the rank-$(W,S)$00 principle “all biclosed sets are weakly separable” persists outside the affine and universal cases (Barkley et al., 2023, Barkley et al., 31 Aug 2025).

The significance of the Coxeter-theoretic conjecture lies in its attempt to repair a basic defect of weak order in infinite type. Ordinary weak order loses joins, whereas the extended weak order enlarges inversion sets to all biclosed sets. The affine and rank-$(W,S)$01 universal results show that, in substantial infinite families, this enlargement restores lattice structure and admits explicit join formulas via closure or $(W,S)$02-closure. A plausible implication is that biclosed sets, weak separation, and hyperplane-arrangement geometry capture a large part of the order-theoretic structure that weak Bruhat order alone loses in infinite Coxeter groups.