Let $\Bbb{Y}$ be Young's lattice which, as you'll recall, is the lattice of all integer partitions ordered by inclusion (of their Young shapes); let $\Bbb{Y}_n$ be the set of all partitions $\lambda \vdash n $, i.e. the set of rank $n$ elements in $\Bbb{Y}$. Stanley observed in his paper "Differential Posets" that for a fixed partition $\mu \vdash n$ the formal linear combination $X_\mu := \sum_{\lambda \vdash n} \chi_\mu^\lambda \, \lambda$ is an eigenfunction of the composite operator $UD_n$ where: (1) $\chi^\lambda_\mu$ is the $S_n$ irreducible character associated to $\lambda \vdash n$ evaluated on the conjugacy class $C_\mu \subset S_n$ corresponding to $\mu \vdash n$ and (2) $U, D: \Bbb{C}\Bbb{Y} \longrightarrow \Bbb{C}\Bbb{Y}$ are the Up and Down operators acting on the vector space $\Bbb{C}\Bbb{Y}$ of all $\Bbb{C}$-linear combinations of partitions, and defined by
\begin{equation} U\lambda := \sum_{\mu \, = \, \lambda + \Box} \mu \quad \qquad D\lambda := \sum_{\lambda \, = \, \mu + \Box} \mu \end{equation}
and where $D_n$ and $U_n$ denote the restrictions of $U$ and $D$ to the finite dimensional subspace $\Bbb{C}\Bbb{Y}_n$.
For two partitions $\lambda, \rho \vdash n$ the product of their characters (evaluated at $\mu \vdash n$) decomposes as
\begin{equation} \chi^\lambda_\mu \, \chi^\rho_\mu = \sum_{\sigma \vdash n} K_{\lambda, \rho}^\sigma \, \chi^\sigma_\mu \end{equation}
for non-negative integers $K^\sigma_{\lambda,\rho}$ known as the Kronecker coefficients. Assemble these Kronecker coefficients into a $\Bbb{Y}_n$-by-$\Bbb{Y}_n$ incidence matrix $A(\rho)$ (of an unoriented graph $\Gamma(\rho)$ with vertices indexed by elements in $\Bbb{Y}_n$) whose matrix entries are given by
\begin{equation} A(\rho)_{\lambda,\sigma} := K^\sigma_{\lambda,\rho} \end{equation}
We may view $A(\rho)$ as an endomorphism of $\Bbb{C}\Bbb{Y}_n$ in which case, by construction, $X_\mu$ will be an eigenfunction of $A(\rho)$ with eigenvalue $\chi^\rho_\mu$.
Question: Is $A(\rho)$ implemented by some (balanced?) polynomial combination of $U$ and $D$ operators ?
Motivating Example: In the case of the partition $\rho = (n-1, 1)$ corresponding to the standard representation, the character values are given by $X^\rho_\mu = m_1(\mu) - 1$ where $m_1(\mu)$ is the number of parts of $\mu \vdash n$ equal to $1$ and so $A(\rho)$ coincides with the operator $UD_n - \Bbb{1}_n$ where $\Bbb{1}_n$ is the identity operator on $\Bbb{C}\Bbb{Y}_n$.