# Step by step procedure to obtain irreducible representations and construct character table of a group

I am studying group theory and character table of $S_2$ is given in the book. But how to obtain this table is not given.

Can someone explain how exactly to construct this table?

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Once you have finished the book, you'll be able to fill in such tables yourself. And for S2, it will take like 30 seconds. – Sjoerd Nov 20 '11 at 16:45
@Pratik: Your accounts will be merged soon; unfortunately there seems to be an issue that the SE developers have to deal with first though. – Zev Chonoles Nov 27 '11 at 22:38

S2 is a group with two elements, call them 1 and t. A representation of S2 takes 1 to the identity matrix and t to some matrix, so we can think of representations X of S2 as simply some matrix X(t). Of course not just any matrix will do, since X is a homomorphism and so $(X(t))^2 = X(t^2) = X(1)$ has to be the identity matrix. In other words, X(t) has to be a matrix whose square is the identity. All of its eigenvalues also have to square to the identity, so they are all ±1. By considering the Jordan canonical form of X(t), you can see that X(t) must be diagonalizable (since a large Jordan block does not square to the identity).
$$\begin{array}{r|rr} S_2 & 1 & t \\ \hline X_1 & 1 & 1 \\ X_2 & 1 & -1 \end{array}$$