bibkey: hivert2026okada authors: Hivert, Florent; Scott, Jeanne year: 2026 title: Diagrammatic Okada monoid and cellularity of the Okada algebra doi: 10.48550/arXiv.2609.01440 url: https://arxiv.org/abs/2609.01440v1 claim: Conjecture 7.9 identifies the regular-trace discriminant with a product of powers of cell-form Gram determinants. strata_touched:
- D5/S0/Certificates/Combinatorics/OkadaDiscriminantRefutation license: citation-only triage: anchor
The Okada discriminant
Conjecture 7.9, page 60, states:
det G_N = ∏_{S ∈ 𝕐𝔽𝕊_N} [det G^S_N]^{2 dim(Fib(S))}.
Its preamble on the same page states:
Recall the discriminant of the Okada algebra is determinant of the N! × N! matrix G_N with entries given by [G_N]_{σ,τ} := tr [ϱ^reg(E_σ)ϱ^reg(E_τ)] where ϱ^reg is the left-regular representation of O_N(X,Y) and σ, τ ∈ S_N.
Definition 3.1, page 10, states:
Fix a positive integer N. Given a field 𝕂, let X = (x₁, …, x_{N−1}) and Y = (y₁, …, y_{N−2}) be two sequences of element of 𝕂. The Okada algebra O_N(X,Y) is the algebra generated by {E_i | i = 1 … N − 1} and subject to the relations E_i² = x_i E_i (1 ≤ i ≤ N − 1), E_i E_j = E_j E_i (|i − j| ≥ 2), E_{i+1} E_i E_{i+1} = y_i E_{i+1} (1 ≤ i ≤ N − 2).
Definition 4.5, page 33, states:
A Fibonacci set of rank N is a subset S = {s₁ < s₂ < ··· < s_k} of [N] whose size k has the same parity as N and such that s_i has the same parity as i for all i ∈ {1, …, k}.
Definition 3.39, page 23, states:
A labelled, non-crossing arc-diagram is called an Okada arc-diagram if the following conditions are satisfied: (1) the label of each arc a — b must be at least 1 and at most min(|a|, |b|), (2) the label of each arc a — b must have the same parity as min(|a|, |b|). (3) if an arc a — b is nested in an arc c — d then the label of a — b is strictly larger than the label of c — d.
Page 31 states:
A half Okada arc-diagram H is a labelled, non-crossing half arc-diagram which satisfies Conditions 1 to 3, or, equivalently, if gluing H with its mirror produces a valid Okada arc-diagram.
Section 6.1 “Restriction and induction of cell modules”, equation (35), page 50, states:
The left O_N(X,Y) cell module V^S associated to S ∈ 𝕐𝔽𝕊_N can be realized by the vector space spanned by rank N half Okada arc-diagrams |D⟩ such that PLab|D⟩ = S equipped with the left action • obtained by extending linearly the formula E_C • |D⟩ := {λ(C,D) |C · D⟩ if PLab(C · D) = S; 0 otherwise} where D ∈ O_N(X,Y).
Definition 7.7, page 60, states:
Let S be a rank N Fibonacci set and let V^S be the associated cell module of O_N(X,Y). Attached to this module is an O_N(X,Y)-invariant bilinear form φ_S: V^S × V^S → 𝕂 implicitly defined for two half diagrams H, K with propagating sets PLab(H) = PLab(K) = S by E_{H ⋈ H} • K = φ_S(H,K) H.
The Gram matrix G^S_N of the bilinear form φ_S is the dim V^S × dim V^S matrix whose entries are given by [G^S_N]_{H,K} := φ_S(H,K).
The encoding uses zero-based generator and arc labels, a quotient of the free algebra by precisely the displayed relations, and the lexicographically minimal words of Definitions 3.3–3.4. Its trace is the trace of left multiplication on that quotient. Half diagrams are finite noncrossing labelled incidence maps; their propagating label sets index the cells. Full diagrams are pairs of halves with equal propagating labels. Recursive flattening translates these diagrams into words in the quotient.
The diagram-coefficient definition takes inverse coordinates when diagram expansion is bijective and is zero otherwise. The action-defined cell form is zero when its scalar equation has no solution. General scalar existence, bijectivity and flattening correctness are not proved in this module. At the rational witness N = 3, X = (1,2), Y = (1), the six quotient words form a basis, all six diagram flattenings are checked, expansion is bijective, and the cell action’s ket independence and defining form equation are checked. The cell form selects the scalar satisfying that equation; its agreement with diagram coordinates is proved before computing the cell Gram matrices. Thus the zero fallback is unused in the refutation.
At this witness the regular Gram determinant is −16. The three cell dimensions are 1, 1, 2 and their Gram matrices are (1), (1), and ((2,1),(1,1)), each with determinant 1. The cell product is consequently 1. These are checked inside the proof of the single settling theorem.
The refutation concerns the exact equality in Conjecture 7.9. It does not refute the separate cell-determinant formula in Conjecture 7.8 or the basis and cellularity theorems. The discussion of discriminants and meanders in Section 9.4.2 cannot use Conjecture 7.9 as an exact identity. A corrected scalar factor and its characteristic dependence remain separate questions.
Verified locator
- Source: https://arxiv.org/abs/2609.01440v1
- DOI: https://doi.org/10.48550/arXiv.2609.01440