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

  1. Mar 11, 2010 #1
    1. The problem statement, all variables and given/known data
    5.4: If sigma in S_n has cycle type n_1,...,n_r, what is sgn(sig)? (sgn is the sign homomorphism)


    2. Relevant 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.


    3. 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.
     
  2. jcsd
  3. Mar 12, 2010 #2

    Dick

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    I'm not quite following your notation here. But to get the sign you just take (-1)^(number of even length cycles), right?
     
  4. Mar 12, 2010 #3
    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.
     
  5. Mar 12, 2010 #4

    Dick

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    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.
     
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