If a given permutation in S_n has a given cycle type, describe sgn(sig).

[Edellaine]

Homework Statement

5.4: If sigma in S_n has cycle type n_1,...,n_r, what is sgn(sig)? (sgn is the sign homomorphism)

Homework Equations

sgn(sigma) = 1 if sigma is even. sgn(sigma) = -1 is sigma is odd
cycle type is the length of the cycle type. If n_2 = 2, sigma has two 2-cycles.

The Attempt at a Solution

I know what cycle type is (if n_i = j, there are j cycles of length i), and sgn(sigma) is easy. How would I go about expanding (sig) to find how many two cycles I have, if that's even what I should be doing. It doesn't seem that worthwhile to generalize this for cycle type.

I was thinking of writing sgn(sig) as a product of sgn(sig_i) where (sig_i) is an individual cycle of the product of cycles forming (sig), but I don't think that exactly accounts for multiples.

I'm also not sure how to come up with conditions that will say, depending on r, if sgn(sig) = 1, or sgn(sig) = -1.

 

[Dick]

I'm not quite following your notation here. But to get the sign you just take (-1)^(number of even length cycles), right?

 

[Edellaine]

Yup. But since you can write any n-cycle as a product of 2-cycles, how do I account for the cycles of odd length.

 

[Dick]

Cycles of even length break into an odd number of two cycles, cycles of odd length break into an even number. I don't see the problem.