#Permutation math series

One of the families of finite simple groups are the alternating groups, the group of even permutations on a set with at least five elements. Incidentally, I mentioned even and odd permutations a few days ago in the context of finite simple groups. Two different ways of getting from one arrangement to another may use a different number of swaps, but the number of swaps in both approaches will have the same parity.)

#Permutation math series

You could change one order of indices to another by different series of swaps. (There’s an implicit theorem here saying that the definition above makes sense.

You could think of putting the indices into a bubble sort algorithm and counting whether the algorithm does an even or odd number of swaps. If it takes an odd number of swaps, the sign is -1. If you can order the indices with an even number of swaps, the sign of the permutation is 1. Otherwise, the symbol evaluates to 1 or -1.

If any two of the subscripts are equal, the symbol evaluates to 0. The permutation symbol, sometimes called the Levi-Civita symbol, can have any number of subscripts. This is analogous to how you might reduce the complexity of a computer program. In other words, the symbol encapsulates some complexity, keeping it out of your calculation. It has some branching logic built into its definition, which keeps branching out of your calculation, letting you handle things more uniformly. That is, you don’t have to write “if … else …” in when doing your calculation. Because branching logic is built into the symbol, it can keep branching logic outside of your calculation. One example of this is the Kronecker delta function δ ij which is defined to be 1 if i = j and zero otherwise. Sometimes simple notation can make a big difference.