Elementary transposition

Now we may find the matrix representation, U, of the operators. The dimensions of the matrices will be the same as the dimensions of the irreducible representation used. The matrix representation of the identity operator, U E), will of course be the identity matrix. If it is noted that any permutation may be written as a product of transpositions (permutations of order 2), and any transposition may be written as a product of elementary transpositions p p + 1) [74], then it is only nessesary to find matrix representations of the elementary transpositions. The diagonal elements of the elementary transposition p p + 1) are given by... 

If we wanted to generate the representation matrices of 4, we would find the three elementary transpositions and the identity and then generate the other 20 matrices. On the other hand, we could find the representations of the group S2 in 54. This consists of two matrices, U[ E)] and C/[(12)]. We could then use C/[(23)] and f/[(13)] to generate all of the six elements in S3. We could then use C/[(34)j, f/[(14)], and f/[(24)] to generate the rest of 54. While this may seem at first more time-consuming, it is much more easily automated than the brute force approach. 

A transposition (k — 1, k) which interchanges two adjacent elements is called an elementary transposition. When this elementary transposition acts on a branching diagram spin function QfM, it affects only the arcs corresponding to k — 1 and k, that is the path segments contained between... 

The elementary permutations are the identity that leaves unchanged the i rd of any determinant, and the permutations that cause a single pairwise transposition in the order of the spin orbitals. The sign of the elementary permutations is given by (— 1/, where t is the number of pairwise transpositions. For the diagonal element in Equation 3.A.3 the elementary permutations are,... 

Successive applications of the elementary permutations are used to construct more complex permutations. In this manner, we generate composite permutations that perform two, three, or more, pairwise transpositions within each spin—orbital subset, or composite permutations which perform the transpositions simultaneously on the two spin—orbital subsets. For our example, the... 

When a stochastic model is described by a continuous polystochastic process, the numerical transposition can be derived by the classical procedure that change the derivates to their discrete numerical expressions related with a space discretisation of the variables. An indirect method can be used with the recursion equations, which give the links between the elementary states of the process. 

Before discussing size-extensivity, we consider the commutation relations between two operators Op and Ob associated with fragments A and B, respectively. We first assume that both operators represent a single string of elementary operators and recall that the elementary operators of Oa anticommute with those of Ob. The operators Op and Ob therefore anticommute if an odd number of transpositions of elementary operators is required to reorder Op and Ob and otherwise commute. The number of transpositions is equal to the number of elementary operators in Op times the number of elementary operators in Ob. Therefore, Op and Ob anticommute if both operators contain an odd number of elementary operators and otherwise commute ... 

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