1. Notations for permutations

There are many ways to encode a permutation of several objects. This package uses four of them. For example suppose that the collection of six objects
    ♡    ⌚    □    ○    :-)    △
is reordered to
    □    ⌚    ○    ♡    △    :-)
1) We can say that the first object (a heart) has gone to the fourth place (it is conventional to say that the "image" of "point" 1 is point 4), and so on. If we number the initial objects from 1 to 6, one natural way of encoding this permutation would be
    Perm[{3, 2, 4, 1, 6, 5}]
Note that the point 2 is not moved by the permutation. It is called a "singleton".
2) Another natural way of encoding the permutation would be just listing the images of the six points:
    Images[{4, 2, 1, 3, 6, 5}]
3) It is well known that any permutation can be understood as a set of disjoint cycles where the image of each point is written on its right and the image of the last point of the cycle is the first one:
    Cycles[{1, 4, 3}, {5, 6}]
By convention, singleton cycles like {2} are removed from the cyclic notation.
4) Finally, it is particularly efficient in Mathematica to represent permutations using rules:
    Rules[1->4, 4->3, 3->1, 5->6, 6->5]
Again, note that the singleton has not been included.

Perm[list]                Images[list]
Cycles[cycles]            Rules[rules]

Notations for permutations.

Naturally, permutations form a group under the law of multiplication. For example all permutations of n points form the so-called symmetric group S_n. However there is another important superstructure that will be very useful when working with tensor symmetries: an algebra. We shall define laws of addition of permutations and of multiplication by scalars. At the moment we only assume that permutations can be given a sign (-1 or +1). It that sense, following R. Portugal, we are working on {-1,1}×S_n. An example of signed permutation is:
    - Perm[{2, 3, 4, 1}]

We can translate from one notation to other notation:


TranslatePerm[Perm[{3, 2, 4, 1, 6, 5}], Images]


Images[{4, 2, 1, 3, 6, 5}]


TranslatePerm[%, Cycles]


Cycles[{1, 4, 3}, {5, 6}]


TranslatePerm[%, Rules]


Rules[1→4, 4→3, 3→1, 5→6, 6→5]


TranslatePerm[%, Perm]


Perm[{3, 2, 4, 1, 6, 5}]

The sign just remains untouched:


TranslatePerm[-Perm[{3, 2, 4, 1, 6, 5}], Images]


-Images[{4, 2, 1, 3, 6, 5}]

The notation of a permutation is given by


NotationOfPerm[Cycles[{1, 4, 3}, {5, 6}]]



TranslatePerm        Change permutation to a given notation
NotationOfPerm        Returns the notation of a permutation

Translation and notation of permutations.

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