A handout that deals with some of the concepts on this page (including permutations, sign, and cycle decompositions) is here (PDF).
Permutations and Cycle Decompositions
This video provides a review of ways we can represent permutations that relies on a geometric approach. We also introduce cycle decompositions give us a way of representing permutations.
Composing Permutations
The order in which we compose permutations matter. This video describes a geometric model for combining them by using σ and τ and making a connection to geoboards.
The Sign of a Permutation
We can define the sign of a permutation as the number of crossings in the geometric representations we have been using. We will calculate the sign of the permutations σ and τ as examples.
Verifying That sgn(τσ) = sgn(τ)sgn(σ)
By looking at sgn(τσ) we can develop an understanding of why sgn(τσ) = sgn(τ)sgn(σ).
The Sign of a Cycle
One last result that we will need on permutations. If we can figure out the sign of a single cycle, we can figure out the sign of any permutation using our formula sgn(τσ) = sgn(τ)sgn(σ). This video introduce a formula that we can use for finding the sign of a cycle, by first looking at the sign of a 2-cycle, which is referred to as a transposition.
Proof for the Sign of an l-cycle
This video provides a sketch of how one would prove that the sign of an l-cycle is (-1)l-1.
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