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Unformatted text preview: Math 220C May 9, 2008 Further examples of group representations In class, we discussed several additional examples of group representations. (1) Present Z n as h x | x n = 1 i , and let C be a primitive n th root of unity. Consider the matrix representation T : Z n GL(2 , C ) defined by T ( x ) = - 1 . This is a direct sum of two one-dimensional representations. (2) Consider a group BD 2 m presented as BD 2 m = h x,y | x 2 m = 1 ,y 2 = x m ,y- 1 xy = x- 1 i . (Note that the subgroup generated by x is isomorphic to Z 2 m .) Let C be a primitive 2 m th root of unity, and define a matrix representation T : BD 2 m GL(2 , C ) by T ( x ) = - 1 T ( y ) = 1- 1 0 . (Note that when restricted to the subgroup Z 2 m generated by x , this is the same representation defined in example 1. (3) Consider a group BT presented as BT = h x,y,z | x 4 = 1 ,y 2 = x 2 ,z 3- 1 ,y- 1 xy = x- 1 ,z- 1 xz = xy,z- 1 yz = x- 1 i ....
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