A plethystic chain rule

Abstract

We consider a derivation $\mathsf{D}$ on the ring $\hat{I}›$ 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 $\hat{I}›$ 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 $f\in \hat{I}›$ is an homogeneous symmetric function, to that of $f$.

18 pages. Comments are welcome. v2: title changed and introduction updated. v3: major changes to the text, amongst which a connection between the derivation and the character theory of the symmetric group, and an interpretation of our chain rule via Adams operations; main results unchanged