Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules

TL;DR

This paper develops a differential-operator framework for representing the symmetric group $S_n$ and identifying singular vectors in Verma modules of $sl(n,b{C})$, connecting classical results with PDE methods.

Contribution

It introduces a PDE-based approach to characterize singular vectors and provides a differential-operator representation of $S_n$ on truncated power series spaces.

Findings

Solution space spanned by symmetric group actions on 1

Singular vectors characterized as polynomial solutions

Unified approach encompassing classical results

Abstract

Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group $S_n$ on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by $\{\sgm(1)\mid \sgm\in S_n\}$. Moreover, the singular vectors of $sl(n,\mbb{C})$ in the Verma module are given by those $\sgm(1)$ that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\mbb{C})$ are naturally included in our almost elementary approach of partial differential equations.