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

Simple item page
dc.creatorXu, Xiaoping
dc.date2009-03-25
dc.date.accessioned2026-07-07T12:56:21Z
dc.date.available2026-07-07T12:56:21Z
dc.descriptionGiven 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.
dc.description22pages; This is a reformulation of our earlier manuscript "Partial Differential Equations for Singular Vectors of sl(n)" (arXiv:math/0305180)
dc.identifierhttps://arxiv.org/abs/0903.4239
dc.identifierhttp://arxiv.org/abs/0903.4239
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/224552
dc.subjectRepresentation Theory
dc.subjectQuantum Algebra
dc.subject17B10; 17B20; 35C05
dc.titleDifferential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules
dc.typetext

Files

Collections

arXiv