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

2009/03/25 by Xiaoping Xu, Xu, Xiaoping
Mathematics · Physics and Astronomy · #17B10 #17B20 #35C05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0903.4239

openalex publication_date 2009/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a weight of sl(n,\mbbC), 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)| \sgm∈ S_n\. Moreover, the singular vectors of sl(n,\mbbC) 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,\mbbC) are naturally included in our almost elementary approach of partial differential equations.

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