MHB Proving Evenness of m and r in σ ∈ Sn

  • Thread starter Thread starter namzay300
  • Start date Start date
namzay300
Messages
2
Reaction score
0
Suppose σ ∈ Sn and σ = τ12****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!
 
Physics news on Phys.org
namzay300 said:
Suppose σ ∈ Sn and σ = τ12****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.
 
Thread 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...

Similar threads

Replies
2
Views
5K
Back
Top