I have a permutation that has been expressed in disjoint cycles (this isn't my actual question, this is an example done in lectures which I'm trying to understand):
(a b c)(d e f g h i)
Now the procedure taught to calculate the sign of the permutation is to multiply the signs of the disjoint cycles, but in the example it is written:
$ (-1)^2 \cdot (-1)^5 = -1 $
Why is it being raised to the powers of 2 and 5? Should it not be 3 and 6 because that is the length of the cycles?
Because when I try my question, I calculate the inversions of the permutation and using that gives me the sign to be negative, but the disjoint cycle has 8 "elements" so $ (-1)^8 = 1 $, which is not negative.
So does that mean you always do $(-1)^{\mathrm{(Length\, \, of\, \, cycle\, \, – 1)}} $ and then multiply like this?
Best Answer
The sign of a cycle of length $n$ is indeed $(-1)^{n-1}$. That is because transpositions have sign $-1$ and permutations of length $n$ can easily be rewritten as the product of $n-1$ transpositions by just splitting them up. For example $(a b c) = (a b) (b c)$.