Structure Theorems for the Symmetric Groups Acting on its Natural Module

2013/01/05 by Robert Mckemey, Mckemey, Robert
Mathematics · #16W22 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16W22

paper · pdf · doi:10.48550/arxiv.1301.0947

12 pages

arxiv created 2013/01/05 · openalex publication_date 2013/01/05 · arxiv updated 2013/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper gives an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module. Let G be the symmetric group on x₁,..., x_n and let d_i be the i^th elementary symmetric polynomial in the x_i's. We show that if we take monomial representations discussed in \cite[Section 3]Kemper to be the modules V_I, then we have an isomorphism of kG-modules k[x₁,..., x_n] ≅ \Oplus_\n\ ⊆ I ⊆ [n] k[d_I] ⊗_k V_I.

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