2006/09/09 by Hanspeter Kraft, Kraft, Hanspeter, Gerald W. Schwarz +1
Computer Science · #14L30 #14R20 #20C15 #20G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0609253
openalex publication_date 2006/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite group and ϕ\colon V→ W an equivariant morphism of finite dimensional G-modules. We say that ϕ is faithful if G acts faithfully on ϕ(V). The covariant dimension of G is the minimum of the dimension of ϕ(V) taken over all faithful ϕ. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree n-3 for a generic field extension of degree n≥ 5.