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Linear representation theory of symmetric group:S3

8 bytes added, 03:01, 12 October 2011
Standard representation
Since the representation is realized over <math>\mathbb{Z}</math>, it makes sense over all characteristics. The only characteristic where it is not irreducible is characteristic 3. In characteristic 3, the representation is indecomposable but not irreducible.
Here is an alternative perspective on this representation in characteristic 3. The symmetric group is identified with the [[GA(1,q)|general affine group of degreeone]] one over the [[field:F3|field of three elements]]. In other words, it is the semidirect product of the additive group of this field (a [[cyclic group:Z3|cyclic group of order three]]) and the [[cyclic group:Z2|multiplicative group of this field]], where the multiplicative group acts on the additive group by multiplication. Via the general fact that embeds a general affine group in a general linear group of one size higher, we get a faithful representation of the symmetric group on three elements in the [[general linear group of degree two]] over [[field:F3]], i.e., in <math>GL(2,3)</math>.
==Degrees of irreducible representations==
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