MHB Proving Evenness of m and r in σ ∈ Sn

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The discussion focuses on proving that for a permutation σ in Sn expressed as a product of transpositions, the number of transpositions m is even if and only if another number r is even. It highlights the relationship between the action of transpositions on the Vandermonde polynomial and how this affects the sign of the polynomial. Specifically, if σ is an even permutation, it does not change the sign of the Vandermonde polynomial, while an odd permutation does. The argument concludes that if a permutation could be both even and odd, it would lead to a contradiction, reinforcing the evenness condition. This establishes a crucial link between the parity of m and r in the context of permutations.
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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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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)
 
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.
 
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.
 
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