Order dimension versus rank for Coxeter weak orders
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Must the order dimension of the weak order of a finite Coxeter group equal its rank? The question concerns the minimum number of linear orders whose intersection gives the weak order, including region posets of simplicial arrangements.
Why it matters and what counts as progress
Why it matters: A dimension greater than rank would limit a proposed relation between Coxeter geometry and the order complexity of its weak order.
What counts as progress: Prove equality for new types or certify an obstruction giving dimension greater than rank, including crystallographic types.
How the problem developed
- 2003 · Dimension bounds [1]
Nathan Reading Computed rank-equal order dimensions for weak orders of types A, B and D and gave bounds for other finite Coxeter types.
- 2016 · Conjecture [2]
Nathan Reading Recorded the conjecture that the order dimension of a simplicial arrangement’s region poset equals its rank.
- 14 Aug 2026 · Counterexample [3]
Daria Poliakova; ChatGPT 5.6 Sol Ultra Constructed counterexamples showing dim W(H₄)≥5 and dim W(E₆)≥7, so region-poset dimension can exceed rank. The paper credits the model with finding the obstruction subgraphs.
AI contribution
Order dimension beyond rank for simplicial hyperplane arrangements
Poliakova uses ChatGPT-found finite obstruction subgraphs to show dim W(H₄)≥5>4 and dim W(E₆)≥7>6 for weak orders of Coxeter groups. These region posets of simplicial arrangements disprove the proposed equality of order dimension and rank.
Sources
- Nathan Reading, The Order Dimension of the Poset of Regions in a Hyperplane Arrangement (2003)
- Nathan Reading, Cluster algebras and the order dimension of the poset of regions (2016 chapter)
- Daria Poliakova, Order dimension beyond rank for simplicial hyperplane arrangements (2026)
- Poliakova, order dimension beyond Coxeter rank
Website timeline
Publication dates and editorial changes.
ChatGPT-found examples disprove order-dimension rank equality