Higher Specht polynomials and modules over the Weyl algebra

TL;DR

This paper analyzes the decomposition of certain D-modules related to finite maps and symmetric groups, explicitly constructing simple components using higher Specht polynomials and exploring their invariants and differential operators.

Contribution

It introduces a new explicit construction of simple components of D-modules over finite maps using higher Specht polynomials, advancing understanding of their structure.

Findings

Explicit generators and multiplicities of simple components are provided.

Decomposition of polynomial rings localized at discriminants is described.

Action invariants and differential operators on higher Specht polynomials are studied.

Abstract

In this paper, we study an irreducible decomposition structure of the $\Dc$-module direct image $\pi_+(\Oc_{ \bC^n})$ for the finite map $\pi: \bC^n \to \bC^n/ ({\Sc_{n_1}\times \cdots \times \Sc_{n_r}}).$ We explicitly construct the simple component of $\pi_+(\Oc_{\bC^n})$ by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a $\D$-module decomposition of the polynomial ring localized at the discriminant of $\pi$. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials.