HW5_soln - PHY 5667 Problem Set no 5 solution Problem 1 The...

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PHY 5667 Problem Set no. 5 solution Problem 1 The multiplication table of S 3 (with the convention g table = g A g B ) reads: g A \ g B 1 g 12 g 23 g 31 g 123 g 132 1 1 g 12 g 23 g 31 g 123 g 132 g 12 g 12 1 g 123 g 132 g 23 g 31 g 23 g 23 g 132 1 g 123 g 31 g 12 g 31 g 31 g 123 g 132 1 g 12 g 23 g 123 g 123 g 31 g 12 g 23 g 132 1 g 132 g 132 g 23 g 31 g 12 1 g 123 When talking about permutations of three elements ABC under S 3 , actu- ally all one has to do is to specify the position of two elements (say A and B) after the S 3 -transformation, since the third element (C) then will automati- cally be in the one position not occupied by the first two. The mathematical interpretation is to see S 3 as the surface of x 1 + x 2 + x 3 = 0, where after specifying x 1 , x 2 we automatically know x 3 = - x 1 - x 2 . It suffices to define the S 3 transformations in the basis of { x 1 ,x 2 } , i.e. by 2 × 2 matrices. The group elements in this space are: 1 = parenleftbigg 1 0 0 1 parenrightbigg x 1 = x 1 x 2 = x 2 x 3 = - ( x 1 + x 2 ) = - ( x 1 + x 2 ) = x 3 g 12 = parenleftbigg 0 1 1 0 parenrightbigg x 1 = x 2 x 2 = x 1 x 3 = - ( x 1 + x 2 ) = - ( x 2 + x 1 ) = x 3 g 23 = parenleftbigg 1 0 - 1 - 1 parenrightbigg x 1 = x 1 x 2 = - ( x 1 + x 2 ) = x 3 x 3 = - ( x 1 + x 2 ) = - ( x 1 + x 3 ) = x 2 g 31 = parenleftbigg - 1 - 1 0 1 parenrightbigg x 1 = - ( x 1 + x 2 ) = x 3 x 2 = x 2 x 3 = - ( x 1 + x 2 ) = - ( x 3 + x 2 ) = x 1 1
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g 123 = parenleftbigg - 1 - 1 1 0 parenrightbigg x 1 = - ( x 1 + x 2 ) = x 3 x 2 = x 1 x
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