Absolute-order topology, endpoint conventions, and an exact rank-two family

Definitions and mathematical statement

Let $W$ be a finite Coxeter group and $T$ its set of reflections. Write $\ell_T$ for reflection length, and define absolute order by \[ u\le_Tv\quad\Longleftrightarrow\quad \ell_T(v)=\ell_T(u)+\ell_T(u^{-1}v). \] Its rank is $\ell_T$, with minimum the identity $1$. V. Reiner's source question asks for its topology, EL-labelings/shellings in types $A$ and $B$, and whether passing to the ideal generated by parabolic Coxeter elements repairs the reported type-$D$ obstruction. A parabolic Coxeter element is a Coxeter element in a conjugate of a standard parabolic subgroup, with the intended Coxeter-system convention fixed. The remark asks whether Brady--Watt methods for noncrossing intervals generalize.

For a finite poset $P$, its order complex $\Delta(P)$ has finite chains as simplices. Literal $\Delta(\operatorname{Abs}(W))$ is a cone at $1$, hence contractible. The substantive convention in the aid is \[ K(W)=\Delta(\operatorname{Abs}(W)\setminus\{1\}). \] Deleting a minimum, deleting additional endpoints, and adjoining a maximum give different complexes. An EL-labeling labels cover edges in a total order so each interval has a unique increasing maximal chain, lexicographically first. The source does not fully fix its global labeling convention for this generally nonbounded poset.

The aid reports known homotopy Cohen--Macaulayness for the deleted type-$A$ complex and for the type-$B$ parabolic-Coxeter ideal, EL-shellability of noncrossing intervals, and a non-Cohen--Macaulay interval obstruction in $D_4$. Full hypotheses and references are absent; interval results do not automatically provide a global EL-labeling.

For the dihedral system $I_2(m)$, $m\ge3$, the aid supplies an exact family. There are $m$ reflections of rank one and $m-1$ nonidentity rotations of rank two, every reflection below every such rotation. Therefore \[ K(I_2(m))=K_{m,m-1}\simeq \bigvee^{(m-1)(m-2)}S^1, \] where $K_{a,b}$ denotes the complete bipartite graph. For the ideal generated by parabolic Coxeter elements, only the two full-rank Coxeter rotations remain above the reflections, yielding \[ \Delta(\mathcal J(I_2(m))\setminus\{1\})=K_{m,2} \simeq\bigvee^{m-1}S^1. \] The count uses standard parabolic Coxeter elements, not every arbitrary generating pair of reflections with a newly chosen Coxeter structure.

The aid also reports the product homotopy reduction \[ K(W_1\times W_2)\simeq K(W_1)*K(W_2), \] with the usual join/empty-complex conventions. In particular, for $q\ge1$ and $m_j\ge3$, \[ K\!\left(\prod_{j=1}^qI_2(m_j)\right) \simeq\bigvee^{\prod_j(m_j-1)(m_j-2)}S^{2q-1}. \] These exact rank-two and product statements do not settle the source's general type-$A,B$ EL-labeling or uniform parabolic-ideal program.

Short English statement

Keeping the identity makes the order complex contractible, so the aid studies the complex with that minimum deleted. It computes dihedral examples and a product reduction, while retaining the distinction between known interval shellings and the source's global topology and labeling questions.

Sources

• Local UnsolvedMath registry, permanent ID 20002711, source code catalogue entry; created and updated 14 August 2026. Canonical aim-probability-notes.json, zero-based notes[152], tag problem, background item 3, attributed in the extract to V. Reiner.
• AIM "Braid groups, clusters and free probability": https://aimath.org/WWN/braidgroups/braidgroups.pdf. The exact metadata label accompanies an extraction beginning "Problem 3." followed by "1."; supplied verification says a section heading is merged and symbols damaged. No unverified renumbering is imposed here.
• The original remark cites Brady and Watt [14] for noncrossing interval EL-labeling methods, without a full bibliography. The literature field is empty. Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low; complete citations for the later topology assessments are absent.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.