Proving Evenness of m and r in σ ∈ Sn

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In summary: So this cannot happen, which means the number of permutations in a decomposition must be either all even or all odd. Thus, $m$ and $r$ must have the same parity, and so $m$ is even if and only if $r$ is even. In summary, if a permutation can be written as a product of transpositions, the number of transpositions is even if and only if the number of permutations is even.
  • #1
namzay300
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Suppose σ ∈ Sn and σ = τ1*τ2****τm = t1*t2****tr where each τi and tj is a transposition. Then, prove m is even iff r is even.

Note: τ(δ(r1,r2,...,rn) = -δ(r1,r2,...,rn)

Usually like to provide what I have done so far, but I've been racking my brain for awhile and can't come up with much. Any detailed insight would be much appreciated. Thanks!
 
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  • #2
namzay300 said:
Suppose σ ∈ Sn and σ = τ1*τ2****τm = t1*t2****tr where each τi and tj is a transposition. Then, prove m is even iff r is even.

Note: τ(δ(r1,r2,...,rn) = -δ(r1,r2,...,rn)

Usually like to provide what I have done so far, but I've been racking my brain for awhile and can't come up with much. Any detailed insight would be much appreciated. Thanks!

http://mathhelpboards.com/linear-abstract-algebra-14/symmetric-polynomials-involving-discriminant-poly-17823.html

Are you two taking the same class? (Wink)
 
  • #3
Deveno said:
http://mathhelpboards.com/linear-abstract-algebra-14/symmetric-polynomials-involving-discriminant-poly-17823.html

Are you two taking the same class? (Wink)

That would be quite a coincidence! I was hoping to receive an answer pertaining to my question though :( If anyone out there can help, that would be great! Thank you.
 
  • #4
namzay300 said:
That would be quite a coincidence! I was hoping to receive an answer pertaining to my question though :( If anyone out there can help, that would be great! Thank you.

The link I posted was to another thread that essentially covers the same ground, although it may not seem like it.

The strategy is to show that if $\sigma = (i\ j)$ is a transposition, then the action of $\sigma$ on the Vandermonde polynomial $\delta(x_1,\dots,x_n) = \prod\limits_{k < m} (x_k - x_m)$ given by:
$\sigma(\delta(x_1,\dots,x_n)) = \delta(x_{\sigma(1)},\dots,x_{\sigma(n)})$ is equal to $-\delta(x_1,\dots,x_n)$.

So, by extension, if a permutation $\tau$ can be written as an even number of permutations, it leaves the Vandermonde polynomial unchanged, and if it can be written as an odd number of permutations, it changes the sign of the Vandermonde polynomial.

Now if a permutation could be written as both an even number AND an odd number of permutations, we would have:

$\delta(x_1,\dots,x_n) = -\delta(x_1,\dots,x_n)$, a contradiction.
 

Related to Proving Evenness of m and r in σ ∈ Sn

1. What is the significance of proving evenness of m and r in σ ∈ Sn?

The evenness of m and r in σ ∈ Sn is crucial in determining whether a permutation is even or odd. This, in turn, helps in understanding the structure and properties of the permutation group Sn.

2. How is the evenness of m and r defined in a permutation?

In a permutation, the evenness of m and r is defined by the number of transpositions required to obtain the permutation. If the number of transpositions is even, the permutation is considered even. If the number of transpositions is odd, the permutation is considered odd.

3. What are the methods used to prove evenness of m and r in σ ∈ Sn?

There are multiple methods that can be used to prove evenness of m and r in σ ∈ Sn. Some of these methods include using the definition of even and odd permutations, using the parity of the cycle structure, and using the sign of the permutation.

4. Can the evenness of m and r be proven for all permutations in the group Sn?

Yes, the evenness of m and r can be proven for all permutations in the group Sn. This is because the group Sn contains all possible permutations of n objects, and each permutation can be classified as either even or odd.

5. What are the practical applications of proving evenness of m and r in σ ∈ Sn?

The concept of even and odd permutations has various applications in mathematics and science, including group theory, number theory, cryptography, and physics. It also helps in solving problems related to symmetry and combinatorics.

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