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rep-examples - Math 220C Further examples of group...

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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 x | x n = 1 , and let ξ C be a primitive n th root of unity. Consider the matrix representation ˆ T : Z n GL(2 , C ) defined by ˆ T ( x ) = ξ 0 0 ξ - 1 . This is a direct sum of two one-dimensional representations. (2) Consider a group BD 2 m presented as BD 2 m = x, y | x 2 m = 1 , y 2 = x m , y - 1 xy = x - 1 . (Note that the subgroup generated by x is isomorphic to Z 2 m .) Let ξ C be a primitive 2 m th root of unity, and define a matrix representation ˆ T : BD 2 m GL(2 , C ) by ˆ T ( x ) = ξ 0 0 ξ - 1 ˆ T ( y ) = 0 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 = 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 . (Note that the subgroup generated by x and y is isomorphic to BD 4 .) Let ε C be a primitive 8 th root of unity. Define a matrix representation ˆ T : BT GL(2 , C
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