Mathematical Methods for Physics and Engineering : A Comprehensive Guide

28.4 PERMUTATION GROUPS

Suppose thatφis the permutation [4 5 3 6 2 1]; then

φ•θ{abcdef}=[453621][256143]{abcdef}

=[453621]{befadc}

={adfceb}

=[146352]{abcdef}.

Written in terms of the permutation notation this result is

[453621][256143]=[146352].

A concept that is very useful for working with permutations is that of decom-

position into cycles. The cycle notation is most easily explained by example. For

the permutationθgiven above:

the 1st object,a, has been replaced by the 2nd,b;

the 2nd object,b, has been replaced by the 5th,e;

the 5th object,e, has been replaced by the 4th,d;

the 4th object,d, has been replaced by the 1st,a.

This brings us back to the beginning of a closed cycle, which is conveniently

represented by the notation (1 2 5 4), in which the successive replacement

positionsareenclosed,insequence,inparentheses.Thus(1254)means2nd

→1st, 5th→2nd, 4th→5th, 1st→4th. It should be noted that the object

initially in the first listed position replaces that in the final position indicated in

the bracket – here ‘a’ is put into the fourth position by the permutation. Clearly

the cycle (5 4 1 2), or any other that involved the same numbers in the same

relative order, would have exactly the same meaning and effect. The remaining

two objects,candf, are interchanged byθor, more formally, are rearranged

according to a cycle of length 2, atransposition, represented by (3 6). Thus the

complete representation (specification) ofθis

θ=(1254)(36).

The positions of objects that are unaltered by a permutation are either placed by

themselves in a pair of parentheses or omitted altogether. The former is recom-

mended as it helps to indicate how many objects are involved – important when

the object in the last position is unchanged, or the permutation is the identity,

which leaves all objects unaltered in position! Thus the identity permutation of

degree 6 is

I= (1)(2)(3)(4)(5)(6),

though in practice it is often shortened to (1).

It will be clear that the cycle representation is unique, to within the internal

absolute ordering of the numbers in each bracket as already noted, and that