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Linear representation theory of symmetric group:S3

500 bytes added, 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]] <math>S_n</math> of degree <math>n</math> || <math>n = 3 </math>, i.e., the group <math>S_3</math> || [[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 <math>\mathbb{Z}</math>, and these give irreducible representations over any field of characteristic not dividing the order of the group.
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| [[dihedral group]] <math>D_{2n}</math> of order <math>2n</math> and degree <math>n</math> || <math>n = 3 </math>, i.e., the dihedral group <math>D_6</math> 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 <math>\mathbb{Z}</math> but not in <math>\mathbb{Z}</matH> itself except for degrees 3,4,6 (orders 6,8,12).
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| [[general affine group of degree one]] <math>GA(1,q)</math> over a [[finite field]] of size <math>q</math>|| <math>q = 3</math>, i.e., [[field:F3]] , so the group is <math>GA(1,3)</math>|| [[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]] <math>GL(2,q)</math> over a [[finite field]] of size <math>q</math> || <math>q = 2</math>, i.e., [[field:F2]] , so the group is <math>GL(2,2)</math>. || [[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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