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 " 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[1] 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

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First we assume that maps are written on the left of their arguments, so that compositions take place from right to left. Let   be the symmetric group[2] on   letters, with compositions computed from right to left.

Imagine a situation in which elements of   act[3] on the “places” (i.e., positions) of something. The places could be vertices of a regular polygon of   sides, the tensor positions of a simple tensor, or even the inputs of a polynomial of   variables. So we have   places, numbered in order from 1 to  , occupied by   objects that we can number  . In short, we can regard our items as a word   of length   in which the position of each element is significant. Now what does it mean to act by “place-permutation” on  ? There are two possible answers:

  1. an element   can move the item in the  th place to the  th place, or
  2. it can do the opposite, moving an item from the  th place to the  th place.

Each of these interpretations of the meaning of an “action” by   (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:

  • For each  , move the item in the  th place to the  th place.
  • For each  , move the item in the  th place to the  th place.
  • For each  , replace the item in the  th position by the one that was in the  th place.

This action may be written as the rule  .

Now if we act on this by another permutation   then we need to first relabel the items by writing  . Then   takes this to   This proves that the action is a left action:  .

Now we consider the second interpretation of the action of  , which is the opposite of the first. The following descriptions of the second interpretation are all equivalent:

  • For each  , move the item in the  th place to the  th place.
  • For each  , move the item in the  th place to the  th place.
  • For each  , replace the item in the  th position by the one that was in the  th place.

This action may be written as the rule  .

In order to act on this by another permutation  , again we first relabel the items by writing  . Then the action of   takes this to   This proves that our second interpretation of the action is a right action:  .

Example

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If   is the 3-cycle   and   is the transposition  , then since we write maps on the left of their arguments we have   Using the first interpretation we have  , the result of which agrees with the action of   on  . So  .

On the other hand, if we use the second interpretation, we have  , the result of which agrees with the action of   on  . So  .

Maps written on the right

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Sometimes people like to write maps on the right[4] 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   by its opposite group  . 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   is now a right action, while the action   is now a left action.

Example

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We let   be the 3-cycle   and   the transposition  , as before. Since we now write maps on the right of their arguments we have   Using the first interpretation we have  , the result of which agrees with the action of   on  . So  .

On the other hand, if we use the second interpretation, we have  , the result of which agrees with the action of   on  . So  .

Summary

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In conclusion, we summarize the four possibilities considered in this article. Here are the four variations:

Rule Type of action
  left action
  right action
  right action
  left action

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.

Notes

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  1. ^ For a readable overview of various diagram algebras generalizing group algebras of symmetric groups, see Halverson and Ram 2005.
  2. ^ See James 1978 for the representation theory of symmetric groups. Weyl 1939, Chapter IV treats the important topic now known as Schur–Weyl duality, which is an important application of the place-permutation action.
  3. ^ Hungerford 1974, Chapter II, Section 4
  4. ^ See e.g., Section 2 of James 1978.

References

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  • Tom Halverson and Arun Ram, "Partition algebras", European J. Combin. 26 (2005), no. 6, 869–921.
  • Thomas Hungerford, Algebra. Springer Lecture Notes 73, Springer-Verlag 1974.
  • Gordon D. James, The Representation Theory of the Symmetric Groups. Lecture Notes in Math. 682 (1978), Springer.
  • Hermann Weyl, The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939.