The Character Table of S_3xZ_3

Point of post: In this post we use our recently developed theory on the irreducible representations (characters) of the products of groups to find the character table of a more formidable group, namely .

Motivation

Let’s put our newly developed theory to work.

 

Character Table for the Quaternions

Point of post: In this post we use the techniques we’ve devoloped to construct the character table for the quaternions.

Motivation

We continue our effort of constructing character tables by focusing now on the Quaternions, . We will once again show how beautiful the theory we’ve devoloped can be by constructing the entire table without actually finding a non-trivial irreducible character.

Character Table of S_3 By Finding the Irreducible Representations

Point of post: In this post we construct the first of a few character tables, namely we construct the character table for .

Motivation

We now start off nice and easy and construct the classic character table for using the techniques from the last post. 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 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 . This post shall be useful to illustrate how beautifully simple the construction of character tables is made by the theory we’ve developed.

Character Tables

Point of post: In this post we discuss the notion of the character table of a finite group and discuss several techinques used in their construction.

Motivation

Some people would consider the fact that I have waited this long to talk about characters tables despicable. Admittedly, character tables are perhaps one of the most ‘practical’ application of representation theory to pure group theory. Namely, the character table shall be a numerical array of numbers (a matrix really) which using our past information about the kernel of a character, the center of a character, etc. to shall tell us most of things one may initally like to know about a group. The interesting part, is the using the relations between characters we’ve previously derived we can often get the character table ‘for free’ in the sense that some of the entries are obvious, and the rest can be ascertained from the orthogonality relations between characters, etc.