Grading of Twisted-Involution Bruhat Posets

Definitions and mathematical statement

Let $(W,S)$ be a Coxeter system and let $\theta:W\to W$ be an automorphism with $\theta^2=\operatorname{id}$. In the restored statement define \[ \mathcal I(\theta)=\{w\in W:\theta(w)=w^{-1}\}, \] the twisted involutions. Equip this set with the order induced from the Bruhat order on $W$. One standard Bruhat-order definition says $u\le v$ when a reduced expression for $v$ contains a subword whose product is a reduced expression for $u$.

The restored question is: for which $(W,S)$ and $\theta$ is $\operatorname{Br}(\mathcal I(\theta))$ graded? Since it has least element $1$, the usual grading asks for a rank function $r:\mathcal I(\theta)\to{\mathbb{N}}_0$ with $r(1)=0$ and $r(v)=r(u)+1$ whenever $v$ covers $u$ in this induced subposet. Covers here are not necessarily covers in the full Bruhat order. No finiteness of $W$ is assumed in the extraction.

There is a substantive source-field discrepancy. The original statement instead defines the twisted identities \[ \iota(\theta)=\{\theta(w^{-1})w:w\in W\}, \] asks when $\operatorname{Br}(\iota(\theta))$ is graded, and additionally asks for the topology of its intervals. These are not the same defining set or the same collection of questions. The supplied verification expressly says that official Problem 4 concerns twisted involutions rather than twisted identities, so the restored grading question is displayed above, but the original identity-set and topology clause are retained as unresolved extraction provenance.

For the original topology clause, the usual meaning would be the topology of the order complex of an open interval, whose simplices are its finite chains. The record does not state open versus closed intervals, homotopy versus homology target, or whether that clause belongs to another source item. It must not be silently transferred to the restored involution set. The source also omits whether $\theta$ preserves the distinguished generating set $S$, an essential convention in standard twisted Coxeter theory. Neither version is certified here as the definitive report formulation.

Short English statement

The restored question asks when the Bruhat subposet of twisted involutions is graded. The original instead concerns twisted identities and interval topology, so the draft preserves this conflict and flags the missing assumption about preserving the Coxeter generators.

Sources

• Local UnsolvedMath registry, permanent ID 30000685, code catalogue entry, "Grading of Twisted-Involution Bruhat Posets." Proposed year 2007; proposer unspecified.
• Geometric and Topological Combinatorics (2007), Oberwolfach Reports, pp. 195--272, DOI 10.4171/owr/2007/04; https://doi.org/10.4171/owr/2007/04. This is the sole supplied literature reference.
• Statement status corrected_verified. The supplied verification says official Problem 4 concerns grading twisted involutions, not twisted identities. This conflicts substantively with the retained original defining set and its extra topology question, and requires source reconciliation.
• Literature triage checked 2026-08-21: open. It generically calls the item an unresolved problem/conjecture/direction and reports no authoritative complete resolution in the supplied check. It does not identify which defining set was assessed. Created 2026-08-25; updated 2026-09-16. Exact formulation and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.