A plethystic chain rule
Alessandro D'Andrea, Enrico Fatighenti, Claudio Onorati
TL;DR
The paper develops a derivation ${\mathsf{D}}=\sum_{n\ge0} n\frac{\partial}{\partial {\mathsf{p}}_n}$ on the ring ${\Lambda}$ of symmetric functions and proves a quasi-isometric relation with respect to the Hall product, enabling a recursive computation of Littlewood–Richardson coefficients. It introduces a plethysm chain rule via an auxiliary ring ${\Lambda}[1^s,2^s,\dots]$ and the operator ${\mathsf{D}}_s$, yielding a practical recursion for plethysm coefficients and providing explicit examples. The authors also establish support properties: if a plethysm ${\mathsf{s}}_\alpha[{\mathsf{s}}_\beta]$ involves partitions confined to a column/row region, then all resulting partitions in the expansion are similarly bounded, with precise bounds on the first part and length of the resulting partitions. An application connects these ideas to geometric questions and Chern shadows, linking combinatorial derivations to geometric invariants of Schur functors. Overall, the work offers a recursive framework to compute plethysm coefficients, together with structural bounds on plethysm supports and geometric interpretations via D-set operations.
Abstract
We consider a derivation $\mathsf{D}$ on the ring $Λ$ 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 $Λ$ 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 $f\in Λ$ is an homogeneous symmetric function, to that of $f$.