No cycles in permutation N how to calculate sgn(N^2)?

In summary, the given conversation discusses the calculation of sgn(N^2), where N is a 2 x n matrix. It is mentioned that N^2 has no cycles and therefore, sgn(N^2) can be calculated using the traditional method as sgn(N^2) = (-1)^0 = 1. It is also noted that the cycles in N^2 are 1-cycles, and the identity element of Sn is used to confirm the correctness of the method.
  • #1
ilyas.h
60
0
N is a 2 x n matrix:

N =

1 2 3 4 ... n-1 n
n n-1 ... 4 3 2 1

then N^2 =

1 2 3 4 ... n-1 n
1 2 3 4 ... n-1 nYou COULD use the theorem: sgn(N^2) = sgn(N)sgn(N)

however, I am asked to find sgn(N^2) by the traditional method: sgn(N^2) = (-1)^([L1 - 1] + [L2 - 1]...) where L represents the respective lengths of each cycle. However, N^2 has no cycles, so I am confused.

How would I go about this? thanks.
 
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  • #2
I think I know:

since there are no cycles, you just have sgn(N^2) = (-1)^0 = 1.

can anyone confirm if my method is correct? thanks
 
  • #3
If that's the N^2 you have, the cycles are 1-cycles... So their parity is + ...
In particular you have the identity element of Sn:
[itex]e=(1)(2)...(n)[/itex]
and [itex]W(e)= (1-1)_{first} + (1-1)_{second} + ... + (1-1)_{n-th} =0 [/itex] and so the parity:

[itex] \delta_P(e) = (-1)^{W(e)} = +1 [/itex]
 

Related to No cycles in permutation N how to calculate sgn(N^2)?

1. What does it mean for a permutation to have no cycles?

A permutation with no cycles means that each element in the permutation remains in its original position. In other words, there are no elements that are moved to a different position in the permutation.

2. How is sgn(N^2) calculated for a permutation with no cycles?

The calculation for sgn(N^2) for a permutation with no cycles is relatively simple. Since there are no elements that are moved to a different position, the sign (positive or negative) of the permutation is determined solely by the number of inversions in the permutation. An inversion occurs when a larger number appears before a smaller number in the permutation. If the number of inversions is even, the sign is positive, and if the number of inversions is odd, the sign is negative.

3. Can a permutation have no cycles and still have a negative sign?

Yes, a permutation with no cycles can still have a negative sign. This occurs when there is an odd number of inversions in the permutation, as discussed in the previous answer.

4. How do I know if a permutation has no cycles?

A permutation can be represented visually as a sequence of numbers, where each number represents the position of an element in the permutation. If the sequence forms a "cycle-free" pattern, where each number appears only once and in ascending order, then the permutation has no cycles. Another way to determine if a permutation has no cycles is by calculating its cycle decomposition, which will result in a single cycle for a permutation with no cycles.

5. Can I use a formula to calculate sgn(N^2) for a permutation with no cycles?

No, there is no specific formula for calculating sgn(N^2) for a permutation with no cycles. However, as mentioned earlier, the sign of the permutation can be determined by the number of inversions in the permutation. So, you can use a formula to calculate the number of inversions in a permutation and then determine the sign of the permutation from that result.

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