A plethystic chain rule

TL;DR

This paper explores a derivation on symmetric functions, revealing its algebraic and geometric properties, including a chain rule for plethysm and its behavior as a quasi-isometry on graded components.

Contribution

It introduces a chain-rule formula for a derivation on symmetric functions with respect to plethysm, linking algebraic and geometric aspects.

Findings

Derivation restricts to a quasi-isometry on graded components.

Provides a chain-rule formula for plethysm operation.

Connects the geometry of Schur functions to derivation properties.

Abstract

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