# Abstract Nonsense

## Permutations (Pt.V Transpositions and the Sign of a Permutation)

Point of post: This post (which will undoubtedly spill over into continuations) will discuss the last topic in this sequence of posts on permutations. Namely, we will be discussing the concepts of permutations and the real goal (for our linear algebraic purposes) the sign of a permutation.

Motivation

Up until this point we’ve found a relatively simple class of permutations, cycles,  for which every permutation can be written as a product of them. We take this one step further and show that every cycle can be written as the product of cycles of length 2, or transpositions. Mathematically the sign of a permutation can be thought of  as the answer to the hypothetical question “if $f(x_1,\cdots,x_n)$ is a function such that the interchanging of any two of it’s arguments results in negative the original (e.g. $f(x_1,\cdots,x_n)=-f(x_n,\cdots,x_1)$). Then, if $\pi\in S_n$ is $f(x_1,\cdots,x_n)=f(x_{\pi(1)},\cdots,f_{\pi(n)})$ or $f(x_1,\cdots,x_n)=-f(x_{\pi(1)},\cdots,f(x_{\pi(n)})$?”. Another way to think about it is this. Suppose that we had a regular $n$-gon $P$ in the plane whose “front” is colored red and whose “back” is colored green. Suppose that the $n$ vertices are labeled $v_1,\cdots,v_n$. Then, we could imagine switching two of the vertices by “flipping” the $P$ over the line perpendicular to the line connecting those two vertices. But, we see that in the process we’ve now exposed the green side of $P$. Switching two more vertices results in another flip so that now the red side is exposed. If we permuted the vertices $v_1,\cdots,v_n$ to the configuration $v_{\pi(1)},\cdots,v_{\pi(n)}$ the sign of the permutation will tell us whether the red or green side of $P$ is showing.

November 7, 2010