Proving Evenness of m and r in σ ∈ Sn

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SUMMARY

The discussion centers on proving that for a permutation σ in the symmetric group Sn, the number of transpositions m is even if and only if the number of transpositions r in an alternate representation is also even. The proof hinges on the properties of the Vandermonde polynomial δ(x1, ..., xn), demonstrating that the action of a transposition τ on δ results in a sign change, thereby establishing a contradiction if a permutation could be expressed as both an even and an odd number of transpositions.

PREREQUISITES
  • Understanding of symmetric groups, specifically Sn
  • Familiarity with transpositions and their properties
  • Knowledge of Vandermonde polynomials and their significance in algebra
  • Basic concepts of even and odd permutations
NEXT STEPS
  • Study the properties of symmetric groups and their representations
  • Learn about the implications of transpositions in group theory
  • Explore the applications of Vandermonde polynomials in combinatorial mathematics
  • Investigate the relationship between permutations and polynomial functions
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Mathematicians, students of abstract algebra, and anyone interested in the properties of symmetric groups and polynomial functions will benefit from this discussion.

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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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