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
\sigma
First we assume that maps are written on the left of their arguments, so that compositions take place from right to left. Let
ak{S}n
n
Imagine a situation in which elements of
ak{S}n
n
n
n
n
n
x1,...,xn
x=x1 … xn
n
x
\sigma\inak{S}n
j
\sigma(j)
\sigma(j)
j
Each of these interpretations of the meaning of an “action” by
\sigma
Consider the first interpretation. The following descriptions are all equivalent ways to describe the rule for the first interpretation of the action:
j
j
\sigma(j)
j
\sigma-1(j)
j
j
j
\sigma-1(j)
This action may be written as the rule
x1 … xn\overset{\sigma}{\longrightarrow}
x | |
\sigma-1(1) |
…
x | |
\sigma-1(n) |
Now if we act on this by another permutation
\tau
y1 … yn=
x | |
\sigma-1(1) |
…
x | |
\sigma-1(n) |
\tau
y | |
\tau-1(1) |
…
y | |
\tau-1(n) |
=
x | |
\sigma-1 \tau-1(1) |
…
x | |
\sigma-1\tau-1(n) |
= x | |
(\tau\sigma)-1(1) |
…
x | |
(\tau\sigma)-1(n) |
.
\tau ⋅ (\sigma ⋅ x)=(\tau\sigma) ⋅ x
Now we consider the second interpretation of the action of
\sigma
j
j
\sigma-1(j)
j
\sigma(j)
j
j
j
\sigma(j)
This action may be written as the rule
x1 … xn\overset{\sigma}{\longrightarrow}x\sigma(1) … x\sigma(n)
In order to act on this by another permutation
\tau
y1 … yn=x\sigma(1) … x\sigma(n)
\tau
y\tau(1) … y\tau(n)=x\sigma … x\sigma =x(\sigma\tau)(1) … x(\sigma\tau)(n).
(x ⋅ \sigma) ⋅ \tau=x ⋅ (\sigma\tau)
If
\sigma=(1,2,3)
1\to2\to3\to1
\tau =(1,3)
1\to3\to1
\sigma\tau=(1,2,3)(1,3)=(2,3), \tau\sigma=(1,3)(1,2,3) =(1,2).
x=x1x2x3 \overset{\sigma}{\longrightarrow}x3x1x2\overset{\tau} {\longrightarrow}x2x1x3
\tau\sigma=(1,2)
x=x1x2x3
\tau ⋅ (\sigma ⋅ x)=(\tau\sigma) ⋅ x
On the other hand, if we use the second interpretation, we have
x=x1x2x3\overset{\sigma}{\longrightarrow}x2x3x1 \overset{\tau}{\longrightarrow}x1x3x2
\sigma\tau=(2,3)
x=x1x2x3
(x ⋅ \sigma) ⋅ \tau=x ⋅ (\sigma\tau)
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
ak{S}n
op | |
ak{S} | |
n |
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
x1 … xn \overset{\sigma}{\longrightarrow}
x | |
1\sigma-1 |
… x | |
n\sigma-1 |
x1 … xn \overset{\sigma}{\longrightarrow}x1\sigma … xn\sigma
We let
\sigma=(1,2,3)
1\to2\to3\to1
\tau =(1,3)
1\to3\to1
\sigma\tau=(1,2,3)(1,3)=(1,2), \tau\sigma=(1,3)(1,2,3)=(2,3).
x=x1x2x3 \overset{\sigma}{\longrightarrow}x3x1x2\overset{\tau} {\longrightarrow}x2x1x3
\sigma\tau=(1,2)
x=x1x2x3
(x ⋅ \sigma) ⋅ \tau=x ⋅ (\sigma\tau)
On the other hand, if we use the second interpretation, we have
x=x1x2x3\overset{\sigma}{\longrightarrow}x2x3x1 \overset{\tau}{\longrightarrow}x1x3x2
\tau\sigma=(2,3)
x=x1x2x3
\tau ⋅ (\sigma ⋅ x)=(\tau\sigma) ⋅ x
In conclusion, we summarize the four possibilities considered in this article. Here are the four variations:
Rule | Type of action | ||||||||
---|---|---|---|---|---|---|---|---|---|
x1 … xn\overset{\sigma}{\longrightarrow}
…
| left action | ||||||||
x1 … xn\overset{\sigma}{\longrightarrow}x\sigma(1) … x\sigma(n) | right action | ||||||||
x1 … xn\overset{\sigma}{\longrightarrow}
…
| right action | ||||||||
x1 … xn\overset{\sigma}{\longrightarrow}x1\sigma … xn\sigma | 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.