A plethystic chain rule

We consider a derivation $\mathsf{D}$ on the ring $\mathrm{\Lambda }$ of symmetric functions and investigate its combinatorial, algebraic and geometric properties. More precisely, we show that $\mathsf{D}$ restricts to a quasi-isometry, with respect to the Hall product, on the graded component of $\mathrm{\Lambda }$ of each positive degree and provide a chain-rule formula with respect to the plethysm operation. Furthermore, we relate the geometry of the Schur functions supporting $\mathsf{D}(f)$, where $f\in \mathrm{\Lambda }$ is an homogeneous symmetric function, to that of $f$.