Frobenius reciprocity on the space of functions invariant under a group action

2020/06/04 by Teerapong Suksumran, Suksumran, Teerapong, Tanakorn Udomworarat +1
Computer Science · Mathematics · #05E10 #05E15 #46C99 #Advanced Algebra and Geometry #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Primary 20C15 #Rings, Modules, and Algebras #Secondary 05E18

paper · pdf · doi:10.48550/arxiv.2006.02653

openalex publication_date 2020/06/04 · openalex created_date 2020/06/12 · openalex updated_date 2026/07/28

Abstract

This article studies connections between group actions and their corresponding vector spaces. Given an action of a group G on a nonempty set X, we examine the space L(X) of scalar-valued functions on X and its fixed subspace: L^G(X) = \f∈ L(X)\colon f(a⋅ x) = f(x) \textrm for all a∈ G, x∈ X\. In particular, we show that L^G(X) is an invariant of the action of G on X. In the case when the action is finite, we compute the dimension of L^G(X) in terms of fixed points of X and prove several prominent results for L^G(X), including Bessel's inequality and Frobenius reciprocity.

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