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1) We see that the distinct isomers of U form the carrIer space for a matrix representation {D(si)} (i == 1, ... ) of the symmetric group Sn which is comprised of the n! permutations of the numbers 1,2,... , n. 11) W2 == (b1V~ . Il···I n + ... + If we denote by Wi and W bMV M . )E. Il···I n Il· .. In 2 the note that there is an irreducible representation of Sn corresponding to each partition n1 n2... 11) upon eliminating the redundant terms, then the appropriate expression 2 of n n2 2 ... such that n1 + n2 + ...
Under a transformation A, we have (see Lomont [1959], p. 23) where X~ln2··· denotes the value of the character of the irreducible representation (n1 n2···) of the symmetric group 5n corresponding to the class , of permutations. 20) is over the classes , of 5n . 9). The number of independent components of a three-dimensional tensor of symmetry class (n1 n2 ... 21 ) are tensors of symmetry classes (3), (21), (21) and (111) respectively. A thorough discussion of tensors of symmetry class (n1 n2...
AN } and by expressions such as where the a1"'" ap are constants. 5) A complete set of nth-order tensors. In §4. 7, we list complete sets of invariant tensors of orders 1,2, ... for the groups D2h , 0h' R3 , 03 and if A is a continuous group. The matrix defining the transformation properties of the 3n components x f x~ ... 1 (i 1,... , in == 1,2,3) under A 11 12 In is referred to as the Kronecker nth power of A. 1) which are invariant under A may then be written as Kronecker nth power of A is given by (tr A)n.