The conjectural Gelfand–Tsetlin theory for b-deformed Jucys–Murphy operators

Let $k$ be a positive integer, let $\mathcal{J}_1,\ldots,\mathcal{J}_k$ be the $b$-deformed Jucys–Murphy operators, and let $\mathfrak{e}_k$ be the identity pair partition. Define

$$\mathcal{X}(k)=\langle\mathcal{J}_1,\ldots,\mathcal{J}_k\rangle\cdot\mathfrak{e}_k.$$

Let $\mathsf{Tab}(k)$ denote the standard Young tableaux with $k$ boxes, let $c_b$ be the $b$-content, and write $\overline{\mathsf{T}}$ for the tableau obtained by deleting the box labelled $k$. The $b$-deformed Jucys–Murphy conjecture. The operators commute on $\mathcal{X}(k)$; there is a basis $\{\mathfrak{w}_{\mathsf{T}}:\mathsf{T}\in\mathsf{Tab}(k)\}$ on which they act diagonally by

$$\mathcal{J}_i\cdot\mathfrak{w}_{\mathsf{T}}=c_b(\mathsf{T}_i)\mathfrak{w}_{\mathsf{T}};$$

for $\mathsf{S}\in\mathsf{Tab}(j)$ with $j\leqslant k$, $\mathfrak{w}_{\mathsf{S}}=\sum_{\mathsf{S}\subseteq\mathsf{T}}\mathfrak{w}_{\mathsf{T}}$; and the vectors satisfy the stated recursive interpolation formula. These assertions are conjectural analogues of the known Jucys–Murphy theory for the symmetric group and would provide a corresponding tableau basis for the deformed operators.