# Abstract Nonsense

## Character Table of S_3 Without Finding Irreducible Characters

Point of post: In this post we construct the character table of $S_3$ without having to actually find the irreducible characters.

Motivation

As was stated in our last post this post shall serve to show how nice the theory we’ve devoloped  can make the construction of character tables. In particular, it is often very unapparent percisely how to construct all the irreducible characters. It is just an odd coincidence that for $S_3$ the irreps are so obvious. So, we shall show that in this post except for the trivial irrep which requires no thought to construct we don’t even need to construct either of the other two characters.

Point of post: In this post we construct the first of a few character tables, namely we construct the character table for $S_3$.
We now start off nice and easy and construct the classic character table for $S_3$ using the techniques from the last post. $S_3$ may be perhaps the easiest charater table to construct, but it will give us a good start to stretch our proverbial legs. In this post though, we find the character table using minimal machinery by actually constructing the irreducible characters of $S_3$ instead of using the techniques in our previous post. This method is, in my opinion, for the purpose of  character table construction, not preferable. Indeed, one must actually come up with representatives from each equivalency class of irreps of $S_3$. This post shall be useful to illustrate how beautifully simple the construction of character tables is made by the theory we’ve developed.