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

In summary, the conversation discusses how to determine the sign of a permutation in S_n based on its cycle type. It is stated that the sign can be found by taking (-1) raised to the power of the number of even length cycles. The conversation also mentions the possibility of representing n-cycles as a product of 2-cycles, but it is not clear how this affects the determination of the sign.
  • #1
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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.
 
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  • #2
I'm not quite following your notation here. But to get the sign you just take (-1)^(number of even length cycles), right?
 
  • #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.
 
  • #4
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.
 

1. What is a permutation in S_n?

A permutation in S_n is a rearrangement of the numbers 1 to n, where each number appears exactly once. It is denoted by σ and is represented as a function that maps each number to its new position.

2. What is a cycle type?

A cycle type in a permutation refers to the number of cycles of each length in the permutation. For example, the permutation (1 2 3)(4)(5 6) has a cycle type of (3,1,2) because it has 3 cycles of length 3, 1 cycle of length 1, and 2 cycles of length 2.

3. What is sgn(sig)?

Sgn(sig) refers to the sign of the permutation, which can be either +1 or -1. It is determined by the number of inversions in the permutation, where an inversion is defined as any pair of elements that are in the wrong order. If the number of inversions is even, the sign is +1, and if it is odd, the sign is -1.

4. How do you describe sgn(sig) for a given permutation with a specific cycle type?

To describe sgn(sig) for a given permutation with a specific cycle type, you need to consider the number of cycles of each length and the number of inversions in the permutation. If the number of cycles of even length is odd, then sgn(sig) will be -1. Otherwise, if the number of cycles of even length is even, then sgn(sig) will be +1.

5. Can sgn(sig) be determined by just looking at the cycle type of a permutation?

Yes, sgn(sig) can be determined by just looking at the cycle type of a permutation. As mentioned before, the number of cycles of even length determines the sign of the permutation. Therefore, if you know the cycle type, you can determine sgn(sig) without having to look at the actual elements of the permutation.

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

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