On the Burnside-Brauer-Steinberg theorem

2014/09/26 by Steinberg, Benjamin · 1 citation
#16G99 #16T10 #20C15 #20M30 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1409.7632

Abstract

A well-known theorem of Burnside says that if ρ is a faithful representation of a finite group G over a field of characteristic 0, then every irreducible representation of G appears as a constituent of a tensor power of ρ. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of ρ takes on r distinct values, then the first r tensor powers of ρ already contain amongst them all of the irreducible representations of G as constituents. In this note we prove the analogue of Brauer's result for finite monoids. We also prove the corresponding result for the symmetric powers of a faithful representation.

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