Polytopality and an exact dihedral model for noncrystallographic associahedra

Definitions and mathematical statement

Problem 6.1 (H. Thomas, A. Zelevinsky) asks for a geometric polytopal realization of the cluster complex $\Delta_W$ for a noncrystallographic finite Coxeter group, and more strongly whether its known complete simplicial cluster fan is polytopal. Here $\Delta_W$ is the flag complex of compatible subsets of almost-positive roots $\Phi_{\geq-1}=\Phi^+\cup(-\Pi)$. Incidences require care: a simple polytope $P$ has cluster complex as the boundary complex of its simplicial polar $P^\circ$, not as its own face lattice. The normal-fan realization is the stronger geometric target.

The aid reports that the 2005 question is solved in the literature: Reading--Speyer identify the bipartite cluster fan linearly with a Cambrian fan; Hohlweg--Lange--Thomas realize finite-type Cambrian fans as normal fans of permutahedron-derived polytopes; and Felikson--Tumarkin--Yildirim give 2025 folding-section constructions including $H_3,H_4,I_2(m)$. These supplied status claims lack full references and must not be confused with a newly established theorem in this draft.

The aid additionally gives an explicit rank-two construction, valid for dihedral $I_2(m)$ with $m\geq3$. Let $u_1,\ldots,u_{m+2}$ be the almost-positive-root rays normalized to unit vectors in cyclic order. Let $\theta_j\in(0,\pi)$ be the counterclockwise angle from $u_j$ to $u_{j+1}$, using cyclic indices. Put \[ Q_m=\operatorname{conv}\{u_j\},\qquad P_m=Q_m^\circ=\{x:\langle u_j,x\rangle\leq1\ \forall j\}. \] The origin is interior, every $u_j$ is a vertex of $Q_m$, its boundary cycle is the cluster complex, and the normal fan of $P_m$ is exactly the given cluster fan. Its vertices and Euclidean metric data are \[ v_j=\frac{u_j+u_{j+1}}{1+\cos\theta_j},\qquad \operatorname{Area}(Q_m)=\frac12\sum_j\sin\theta_j, \] \[ \operatorname{Area}(P_m)=\sum_j\tan(\theta_j/2),\qquad \operatorname{Perimeter}(P_m)=2\sum_j\tan(\theta_j/2),\qquad \operatorname{Perimeter}(Q_m)=2\sum_j\sin(\theta_j/2). \] The normalization to unit rays fixes all metric constants; the combinatorial fan is unchanged by that normalization.

For positive support numbers $h_j$, set $P(h)=\{x:\langle u_j,x\rangle\leq h_j\ \forall j\}$ and let $v_j(h)$ be the unique intersection of the adjacent support lines $\langle u_j,x\rangle=h_j$ and $\langle u_{j+1},x\rangle=h_{j+1}$. The aid's necessary-and-sufficient finite stability test is \[ \langle u_\ell,v_j(h)\rangle<h_\ell \qquad\text{for every }j\text{ and every }\ell\notin\{j,j+1\}. \] These strict inequalities ensure that every adjacent intersection is a simple vertex with exactly the prescribed active normals, hence preserve the full normal fan. Non-strict feasibility alone may permit a coarser degenerate fan. This explicit rank-two polar model is separate from the cited higher-rank constructions and does not by itself build $H_3$ or $H_4$.

Short English statement

The source asks for noncrystallographic cluster polytopes, with the correct dual incidence and preferably the exact cluster normal fan. The aid reports general literature solutions and gives a direct dihedral model: take the convex hull of unit cluster rays and polarize it. Adjacent-line intersections provide explicit vertices and a strict finite test for preserving the fan under support changes.

Sources

• Local UnsolvedMath registry, permanent ID $20002722$, catalogue entry; aim-probability-notes.json, notes[163], extracted source item 6.
• American Institute of Mathematics, Braid groups, clusters and free probability, Problem 6.1 (H. Thomas, A. Zelevinsky), https://aimath.org/WWN/braidgroups/braidgroups.pdf. Supplied verification repairs the split heading and subscript, and notes the preceding dual-associahedron/cyclohedron explanation.
• Supplied literature attributions: Reading--Speyer; Hohlweg--Lange--Thomas; Felikson--Tumarkin--Yildirim (2025). Titles, theorem numbers, publication details and authoritative full citations are absent.
• Research attempt 1: partial result, problem solved at run, rigor label "proved," novelty confidence low. Registry classification solved in literature, status solved, statement status exact. The explicit rank-two metric and support test are supplied through the research aid.