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Could anyone help me understand this concept, if we have $1,2,3$ why are $$123,231,213$$ Considered in cyclic order and $$321,213,132$$ Considered in acyclic order? I am asking this in regards to the Levi-Civita symbol, many thanks in advance

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Is it acyclic or anticyclic? – draks ... Oct 15 '12 at 7:29
@draks: my notes say acyclic, I imagine they mean the same thing? – Freeman Oct 15 '12 at 7:31
IMO, $321,\dots$ should better be called anticyclic, because acyclic points to something like $1243$... the best would be to go along with the even/odd permutation definition given here... – draks ... Oct 15 '12 at 7:40
The final sequence in the first line should be 312 - 213 is currently in both lists. The cyclic ones can be found as components of the repetition 123123 which corresponds to putting the numbers around a circle and going clockwise. The other ones are the counterclockwise sequences. With four or more symbols the situation becomes more complex, but the same idea can be used. – Mark Bennet Oct 15 '12 at 7:45
up vote 2 down vote accepted

We take $123$ to be cyclic (essentially by definition) because the numbers are all in order. If you look at this as a cycle around a circle, then the other sequences that start at different points and continue in the same direction are "cyclic" as well. These are $231$ and $312$, found by taking the first number and putting it at the end. I.e. going around the circle we find $1\to 2\to 3\to 1\to 2$, and the cyclic permutations are in this progression. The anti-cyclic permutations are found by going the other way around the circle: $321,213,$ and $132$, found in a similar way.

In general this isn't good enough. For example, $1234$ is cyclic, and the "rotations" still are just like before, but there are permutations like $1324$ that don't correspond visually to a cycle at all. So what do we do? Well it turns out that we can look at the number of transpositions it takes to get to the sequence from the ordered one. A transposition is the name for swapping two elements. For example:

$$123\text{ is 0 transpositions away from 123}$$ $$213, 321,\text{ and } 132\text{ are 1 transposition away from 123}$$ $$231 \text{ and } 312 \text{ are 2 transpositions away from 123}$$

Note that the anti-cyclic permutations are all $1$ away, while the cyclic ones are $0$ or $2$ away. This generalizes quite nicely: a cyclic permutation is one that is an even number of permutations away from the ordered (i.e identity) permutation, while the anti-cyclic permutations are an odd number of transpositions away.

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Thank you so much! – Freeman Oct 16 '12 at 10:40

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