# Changes

## Linear representation theory of symmetric group:S3

, 03:16, 24 February 2013
Family contexts
! Family name !! Parameter values !! General discussion of linear representation theory of family !! Section in this article !! Comparative note
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| [[symmetric group]] [itex]S_n[/itex] of degree [itex]n[/itex] || [itex]n = 3 [/itex], i.e., the group [itex]S_3[/itex] || [[Family version::linear representation theory of symmetric groups]] || [[#Interpretation as symmetric group]] || For any symmetric group on a finite set, all irreducible linear representations can be realized with entries in [itex]\mathbb{Z}[/itex], and these give irreducible representations over any field of characteristic not dividing the order of the group.
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| [[dihedral group]] [itex]D_{2n}[/itex] of order [itex]2n[/itex] and degree [itex]n[/itex] || [itex]n = 3 [/itex], i.e., the dihedral group [itex]D_6[/itex] of order six || [[Family version::linear representation theory of dihedral groups]] || [[#Interpretation as dihedral group]] || For a dihedral group, the irreducible representations can be realized in a finite extension of [itex]\mathbb{Z}[/itex] but not in [itex]\mathbb{Z}</matH> itself except for degrees 3,4,6 (orders 6,8,12).
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| [[general affine group of degree one]] [itex]GA(1,q)[/itex] over a [[finite field]] of size [itex]q[/itex]|| [itex]q = 3[/itex], i.e., [[field:F3]] , so the group is [itex]GA(1,3)[/itex]|| [[Family version::linear representation theory of general affine group of degree one over a finite field]] || [[#Interpretation as general affine group of degree one]] ||
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| [[general linear group of degree two]] [itex]GL(2,q)[/itex] over a [[finite field]] of size [itex]q[/itex] || [itex]q = 2[/itex], i.e., [[field:F2]] , so the group is [itex]GL(2,2)[/itex]. || [[Family version::linear representation theory of general linear group of degree two over a finite field]] || [[#Interpretation as general linear group of degree two]] ||
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