A split source fragment and the dihedral noncrystallographic H-triangle

Definitions and mathematical statement

Source-fragment identity and underlying question. The aid identifies this record as a page-broken continuation of the final bullet of Problem 1.2, not a separate Problem 1.1: the apparent number belongs to a parenthetical cross-reference. Permanent ID and source code are unchanged. The fragment asks how nonnesting partitions and root order extend to noncrystallographic finite Coxeter types, what essential barriers arise for familiar Lie-algebra and affine-hyperplane constructions, and what objects Chapoton's proposed $H$-triangle counts. Its preceding antecedent "these objects" is not fully supplied. The text mentions Kriloff--Ram [33] and Chapoton [18], without the bibliography entries.

In crystallographic root systems, a standard positive-root order has $\alpha\leq\beta$ when $\beta-\alpha$ is a nonnegative integer combination of simple roots, and nonnesting objects are antichains. Extending this formula or its combinatorial consequences to irrational root coefficients is part of the problem, not a fixed definition in the source. The source's suggested barriers are research questions, not an impossibility theorem for every generalized Lie or affine construction.

Reported uniform dihedral model. For each integer $m\geq3$, define an abstract poset with $m$ positive-root labels: two incomparable simple elements $\alpha_1,\alpha_2$, both below $\beta_1$, and a chain \[ \beta_1<\beta_2<\cdots<\beta_{m-2}. \] This fixes the V-then-chain poset up to isomorphism; it does not claim uniqueness among all possible noncrystallographic root-poset axiom systems. With the convention \[ H(s,t)=\sum_{A\text{ antichain}}s^{|A\cap\{\alpha_1,\alpha_2\}|}t^{|A|}, \] the aid's exact enumerator is \[ H(s,t)=1+2st+(m-2)t+s^2t^2. \] The empty antichain, simple singleton antichains, nonsimple singleton antichains and the two-simple antichain account for all terms.

The rank-two $I_2(m)$ cluster complex is represented by a cycle on $m+2$ vertices, with two adjacent distinguished negative vertices and $m$ positive vertices. Weight a face by $x^{\#\mathrm{positive}}y^{\#\mathrm{negative}}$. The independently counted face polynomial is \[ F(x,y)=1+mx+2y+(m-1)x^2+2xy+y^2. \] Its edges comprise $m-1$ positive-positive, two mixed and one negative-negative edge. The aid reports the exact convention-sensitive Chapoton relation \[ H(s,t)=(1-t)^2F\left(\frac{t}{1-t},\frac{st}{1-t}\right), \] \[ F(x,y)=(1+x)^2H\left(\frac yx,\frac{x}{1+x}\right). \] These are polynomial identities after cancellation; the displayed inverse rational substitution is first evaluated where defined. The $H$ variables count simple roots and total size in the stated order, which is the swap needed relative to the alternative standard convention. This works uniformly across crystallographic and noncrystallographic $I_2(m)$, independently of parity or root-length ratio; it is an abstract enumerative model rather than an identification with every geometric order.

The aid additionally records a positive $H_3$ construction and Cuntz--Stump and Chen--Kriloff $H_4$ obstructions, but gives neither complete references nor the exact root-poset axioms/requirements involved. Those obstructions are not promoted to nonexistence of every possible $H_4$ counting object. The dihedral identity does not resolve the full extension program or the essential-barrier question in all noncrystallographic types.

Short English statement

This is a split continuation of a broader Coxeter-theory problem. The aid gives a complete dihedral antichain/cluster-cycle identity with explicit variable conventions, while preserving the incomplete $H_3$/$H_4$ literature qualifications and the unanswered generalization/barrier questions.

Sources

• Local UnsolvedMath registry, permanent ID $20002702$, catalogue entry, aim-probability-notes.json, notes[143], workshop Braid groups, clusters and free probability: https://aimath.org/WWN/braidgroups/braidgroups.pdf. The aid identifies the fragment as the continuation of Problem 1.2; the source item field reads 1.1.
• Original attributions: Cathy Kriloff and Arun Ram [33] on representation-theoretic noncrystallographic issues; Frédéric Chapoton [18] on the $H$-triangle. Bibliography entries are not supplied.
• Supplied research summary records an $H_3$ construction and Cuntz--Stump and Chen--Kriloff $H_4$ obstructions without full references/axioms, and supplies the dihedral coefficient identity. Attempt 1: context_only, not_a_problem, rigor "proved", low novelty confidence; registry open/OPEN-TRIAGE. These source-fragment labels do not erase the underlying questions. The machine-generated aid is not peer reviewed and requires independent verification before citation; no PDF visit is claimed.