Place-permutation action

In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or "places". Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretations of the meaning of "acting by a permutation formula_1" but these lead to four variations, depending whether maps are written on the left or right of their arguments. The presence of so many variations often leads to confusion. When regarding the group algebra of a symmetric group as a "diagram algebra" it is natural to write maps on the right so as to compute compositions of diagrams from left to right.
Maps written on the left.
First we assume that maps are written on the left of their arguments, so that compositions take place from right to left. Let formula_2 be the symmetric group on formula_3 letters, with compositions computed from right to left.
Imagine a situation in which elements of formula_2 act on the “places” (i.e., positions) of something. The places could be vertices of a regular polygon of formula_3 sides, the tensor positions of a simple tensor, or even the inputs of a polynomial of formula_3 variables. So we have formula_3 places, numbered in order from 1 to formula_3, occupied by formula_3 objects that we can number formula_10. In short, we can regard our items as a "word" formula_11 of length formula_3 in which the position of each element is significant. Now what does it mean to act by “place-permutation” on formula_13? There are two possible answers:
Each of these interpretations of the meaning of an “action” by formula_1 (on the places) is equally natural, and both are widely used by mathematicians. Thus, when encountering an instance of a "place-permutation" action one must take care to determine from the context which interpretation is intended, if the author does not give specific formulas.
Consider the first interpretation. The following descriptions are all equivalent ways to describe the rule for the first interpretation of the action:
This action may be written as the rule formula_29.
Now if we act on this by another permutation formula_30 then we need to first relabel the items by writing formula_31. Then formula_30 takes this to formula_33 This proves that the action is a left action: formula_34.
Now we consider the second interpretation of the action of formula_1, which is the opposite of the first. The following descriptions of the second interpretation are all equivalent:
This action may be written as the rule formula_45.
In order to act on this by another permutation formula_30, again we first relabel the items by writing formula_47. Then the action of formula_30 takes this to formula_49 This proves that our second interpretation of the action is a right action: 
formula_50.
Example.
If formula_51 is the 3-cycle formula_52 and formula_53 is the transposition formula_54, then since we write maps on the left of their arguments we have formula_55 Using the first interpretation we have formula_56, the result of which agrees with the action of formula_57 on formula_58. So formula_59.
On the other hand, if we use the second interpretation, we have formula_60, the result of which agrees with the action of formula_61 on formula_58. So formula_63.
Maps written on the right.
Sometimes people like to write maps on the right of their arguments. This is a convenient convention to adopt when working with symmetric groups as diagram algebras, for instance, since then one may read compositions from left to right instead of from right to left. The question is: how does this affect the two interpretations of the place-permutation action of a symmetric group?
The answer is simple. By writing maps on the right instead of on the left we are reversing the order of composition, so in effect we replace formula_2 by its opposite group formula_65. This is the same group, but the order of compositions is reversed.
Reversing the order of compositions evidently changes left actions into right ones, and vice versa, changes right actions into left ones. This means that our first interpretation becomes a "right" action while the second becomes a "left" one.
In symbols, this means that the action formula_66 is now a right action, while the action formula_67 is now a left action.
Example.
We let formula_51 be the 3-cycle formula_52 and formula_53 the transposition formula_54, as before. Since we now write maps on the right of their arguments we have formula_72 Using the first interpretation we have formula_56, the result of which agrees with the action of formula_74 on formula_58. So formula_63.
On the other hand, if we use the second interpretation, we have formula_60, the result of which agrees with the action of formula_78 on formula_58. So formula_80.
Summary.
In conclusion, we summarize the four possibilities considered in this article. Here are the four variations:
Although there are four variations, there are still only two different ways of acting; the four variations arise from the choice of writing maps on the left or right, a choice which is purely a matter of convention.