Non-Induced Representations of Finite Cyclic Groups

2021/10/15 by Ramanujan Srihari, Srihari, Ramanujan
Mathematics · Social Sciences · #11C08 #20C99 (Primary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Philosophy, Sociology, Political Theory #Representation Theory (math.RT) #math.NT #math.RT #msc:11C08 #msc:20C99

paper · pdf · doi:10.48550/arxiv.2110.07987

8 pages

arxiv created 2021/10/15 · openalex publication_date 2021/10/15 · arxiv updated 2021/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be an algebraically closed field of characteristic 0 and let G be a finite cyclic group of order n. In this note we prove, using induction on the number of prime divisors of n, that R_K(G)/I ≅ ℤ[X]/⟨ Φ_n(X) ⟩ where R_K(G) denotes the ring of K-representations of G and I is the sum of ideals Ind_H^G(R_K(H)) of R_K(G) as H varies over all proper subgroups of G. This gives us an idea of how many representations of G are not induced from representations of a proper subgroup.

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