Affine Representations of the Symmetric Group Sn+1 in GL(n,F)

SummaryMany treatments of the representation theory of the symmetric group Sn rapidly delve into erudite considerations of Young diagrams, tableaux, tabloids, and Specht modules. This is fine and natural for specialists, but it can be daunting for undergraduate students. In this article, we show how to develop the standard representation of Sn using little more than the notion of an affinely independent set of vectors. The simplicity of this approach ensures that snippets of the representation theory of the symmetric group Sn may be readily incorporated into a first or second course on linear or abstract algebra. The construction we describe works with any field F and can provide an interesting exercise in working with less familiar fields like the field of rational functions.