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Unformatted text preview: 3.1. EXTERIOR ALGEBRA 57 3.1.2 Permutations of Tensors Let S p be the symmetric group of order p . Then every permutation S p defines a map : T p T p , which assigns to every tensor T of type (0 , p ) a new tensor ( T ), called a permutation of the tensor T , of type (0 , p ) by: v 1 , . . . , v p ( T )( v 1 , . . . , v p ) = T ( v (1) , . . . , v ( p ) ) . Let ( i 1 , . . . , i p ) be a p-tuple of integers. Then a permutation : Z p Z p defines an action ( i 1 , . . . , i p ) = ( i (1) , . . . , i ( p ) ) . The components of the tensor ( T ) are obtained by the action of the permu- tation on the indices of the tensor T ( T ) i 1 ... i p = T i (1) ... i ( p ) . The symmetrization of the tensor T of the type (0 , p ) is defined by Sym( T ) = 1 p ! X S p ( T ) . The symmetrization is also denoted by parenthesis. The components of the symmetrized tensor Sym( T ) are given by T ( i 1 ... i p ) = 1 p !!...
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