## 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 Table for By Finding the Irreducible Characters*

** **We being by determining the degrees of the irreps for . To do this, we recall that . But, we have that (as for any group) admits the trivial irrep which clearly has degree . Thus, we have that

where the sum runs over all with . We recall next that for every we must have that so that but since this implies that . Thus, with these possible choices of it follows from a simple case analysis of that the only solutions are that with and or and for . But, since we know that the degrees of all the irreps of a group are if and only if the group is abelian we may exlcude this second possibility and conclude that the first must be true. This also tells us that the number of conjugacy classes of is . Thus, we may begin setting up our character table as

where we’ve denoted the character of the trivial irrep as , the equivalence class containing as itself, and the characters corresponding to the two other non-trivial classes in as . Moreover, we were able to fill in the first row since the nature of is known and apparent and the first column since . We next seek to obtain representatives for . To do this we do what’s natural. We pick some element of , say and since this conjugacyclass is distinct from the one represented by we may designate it as representative for . Next, we pick another element, say and since we know that the order of is not two and thus cannot be conjugate to and since it is clearly not conjugate to we may conclude that may serve as a representative for . Thus, our table can be rewritten as

We now wish to find the actual characters and by constructing them. Now, for we’re looking for homomorphisms . Now, the first one that may come to mind is the classic homomorhpism . Now, the only question is whether is irreducible. To do this, it suffices to check our alternative characterization of irreducibility. Indeed:

from where the irreducibility of follows. Thus, we may conclude that is what we have called . Thus, since and we may fill in our table accordingly to get

Thus, it remains to find a character that corresponds to a degree two representation. To do this we recall that and interpret as the symmetries of a planar triangle. We can then correspond to the matrix which gives a flip accross the -axis and to the matrix and extend this to a homomorphism since is a generating set for . Explicitly

Now, to check that this is actually an irrep we use our alternative characterization again. Indeed:

From where the irreducibility of follows. It follows then that . Thus, we may complete our character table as:

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print.

2. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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