Coulter–Do conjecture on refined odd Jucys–Murphy operators for b–Hurwitz numbers

Construct, for each positive integer n, a collection of operators J_1,…,J_n on the Coulter–Do graded vector space V_k over C(b) with distinguished vector e_k such that, when restricted to the subspace Xi_n=⟨J_1,…,J_n⟩·e_k, the following hold: (i) the operators commute pairwise; (ii) there exists a basis of Xi_n that simultaneously diagonalizes all J_i; and (iii) this eigenbasis is labeled by standard Young tableaux of size n and satisfies an explicit recursive construction with respect to n. Establishing these operators would provide a precise Jucys–Murphy formalism underpinning b–Hurwitz numbers via a deformation of odd Jucys–Murphy elements and would refine the Schur and zonal actions.

Background

A central theme of the paper is to relate variants of Hurwitz numbers to actions of the ring of symmetric functions via Jucys–Murphy elements. For complex Hurwitz numbers, the Schur action on the center of C[S_n] is classical, while for purely real problems, a zonal action using odd Jucys–Murphy elements exists.

For the Jack deformation corresponding to b–Hurwitz numbers, a representation–theoretic framework analogous to Jucys–Murphy theory is missing. Coulter and Do proposed a conjectural family of operators acting on a graded space V_k (over C(b)) with a preferred vector e_k, expected to yield commuting, simultaneously diagonalizable operators whose eigenbasis is labeled by standard Young tableaux, thereby inducing a Jack-type action.

This work introduces a CJT–refinement interpolating Schur and zonal actions and gives a partial resolution toward this goal, but the full Coulter–Do conjecture remains open in general.

References

In particular, they provide computational evidence for conjecture, that formulates a precise Jucys--Murphy formalism for $b$--Hurwitz numbers (thus specialising to the purely real case for $b=1$) via a deformation of odd Jucys-Murphy elements of. Namely, they invoke a certain graded vector space $\mathcal V_k$ defined over the field $\mathbb C(b)$ and provided with a special element $\mathfrak e_k$ such that:

For a fixed $n\in \mathbb N$ there exists a collection of operators $\mathcal J_1,\ldots,\mathcal J_n \colon \mathcal V_k \to \mathcal V_k$ such that, being restricted to $\Xi_n = \left< \mathcal J_1,\ldots,\mathcal J_n\right>\cdot \mathfrak e_k$ the following properties hold: \begin{itemize} \item the restrictions of $\mathcal J_k$ for $k = 1,\ldots,n $ on $\Xi_n$ commute pairwise; \item there exists a basis of\, $\Xi_n$ that diagonalizes all the $\mathcal J_k$ for $k = 1,\ldots,n$ simultaneously; \item the vectors of the above-mentioned basis are labelled by standard Young tableau of size $n$ and satisfy an explicit recursive relation, allowing to construct them inductively by $n$. \end{itemize}

— A refined twist on Hurwitz numbers  (2508.06188 - Fesler et al., 8 Aug 2025) in Subsection “Symmetric functions and representation theory” (Introduction); Conjecture (Coulter–Do)