bibkey: coulter2025weingarten authors: Coulter, Xavier; Do, Norman year: 2025 title: From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys–Murphy elements doi: 10.48550/arXiv.2506.04002 url: https://arxiv.org/abs/2506.04002v1 claim: Conjecture 5.4(a) asserts commutation of the b-deformed Jucys–Murphy operators on the cyclic orbit of the identity pair partition. strata_touched:
- D5/S0/Certificates/Combinatorics/DeformedJucysMurphyNoncommutation license: citation-only triage: anchor
Deformed Jucys–Murphy operators
Conjecture 5.4(a), page 34, states:
The 𝒥-operators commute when restricted to 𝒳(k) — that is, 𝒥ₘ 𝒥ₙ (v) = 𝒥ₙ 𝒥ₘ (v) for 1 ⩽ m ⩽ n ⩽ k and for all v ∈ 𝒳(k).
Definition 5.3 on the same page states:
For k a positive integer, let 𝒳(k) = ⟨𝒥₁, 𝒥₂, …, 𝒥ₖ⟩ · 𝔢ₖ ⊆ 𝒱ₖ. That is, 𝒳(k) is the orbit of 𝔢ₖ under the action of the algebra of 𝒥-operators.
Definition 5.1, page 33, states:
For k a positive integer, let 𝒱ₖ = ℂ(b)[𝒫ₖ] be the vector space with basis the set of pair partitions of {1, 2, …, 2k}. Define the b-deformed Jucys–Murphy operators 𝒥₁, 𝒥₂, …, 𝒥ₖ: 𝒱ₖ → 𝒱ₖ by 𝒥ᵢ(m) = ∑_(a=1)^(2i−2) ω^(b)((a 2i−1) · m, m) (a 2i−1) · m, where m ∈ 𝒫ₖ and ω^(b) is the weight function of Definition 4.1. We interpret the formula for i = 1 as 𝒥₁ = 0 and refer to these operators collectively as 𝒥-operators.
Definition 4.1, page 21, specifies the charges and weights:
To each vertex v ∈ Γ(m), assign a charge q(v) ∈ {+, −} such that the vertex with the largest label in each cycle is assigned + and such that each edge in Γ(m) is incident to one vertex with positive change and one vertex with negative charge.
Set ω^(b)(m,n) = 1 if q(i) = q(j) and set ω^(b)(m,n) = b if q(i) ≠ q(j).
The source’s “positive change” is retained literally. In the operator formula the output matching is the first argument of the weight function; its graph supplies the charges. The pair-partition action is relabelling: “the pair {a, b} appears in the pair partition m if and only if the pair {σ(a), σ(b)} appears in the pair partition σ · m”. The identity partition has consecutive pairs.
The formal encoding shifts labels down by one and uses fixed-point-free
involutive partner maps on Fin (2*k). Coefficient functions on this finite
carrier represent the vector space. The scalar field is RatFunc ℂ, with
RatFunc.X as the indeterminate. The cyclic orbit uses Algebra.adjoin,
not the full vector space. The alternating walk implements the charge by
the parity of the first occurrence of the component maximum; its general
correspondence to the graph rule is explained mathematically in the
module and mirror, rather than asserted as a separate Lean theorem.
The refutation uses k = 6, v = J_6 J_6 J_5 J_4 J_3 e_6, and the target
pair partition (1 5 | 2 7 | 3 9 | 4 11 | 6 10 | 8 12). The target
coefficients after J_2 J_4 and J_4 J_2 evaluate at b = 2 to 78 and
81. Polynomial transport through the injective map ℤ[X] → ℂ(b) makes
this a refutation with an indeterminate, rather than just a specialized
scalar example.
The published numerical checks for k ≤ 5 do not decide k = 6. Failure
of commutation obstructs the simultaneous eigenbasis conjectured in
5.4(b). Conjecture 5.7(b) requires an ordering convention for products of
these operators; the paper itself notes that its order-independent
interpretation depends on 5.4(a). This refutation alone does not settle
5.7(a), 5.8, or the already proved ordered identities in Proposition 5.9.
Verified locator
- DOI: https://doi.org/10.48550/arXiv.2506.04002
- Source: https://arxiv.org/abs/2506.04002v1