On Symmetric and Anti-symmetric Partial Differential Operators

TL;DR

This paper investigates the behavior of symmetric partial differential operators acting on anti-symmetric functions, revealing the simplicity of the module over the Weyl algebra of elementary symmetric functions.

Contribution

It demonstrates that the module of anti-symmetric functions is simple over the Weyl algebra of elementary symmetric functions, providing new insights into the structure of symmetric PDOs.

Findings

The module of anti-symmetric functions is simple over the Weyl algebra.

Symmetric PDOs preserve the anti-symmetric structure.

The work advances understanding of the algebraic structure of symmetric differential operators.

Abstract

We study the action of symmetric PDO on the module of anti-symmetric functions in $z_1, \dots, z_k$. We show that over the Weyl algebra of the elementary symmetric functions, this module is simple.