B14.3 The Character Table of

The group has seven conjugacy classes, one for each cycle type:

So it has seven irreducible representations, and the squares of their dimensions add up to . We start from the ones we know.

• and , the only -dimensional representations.
• The standard representation , of dimension , with . It is irreducible, since
• The twist , irreducible of dimension , with . It is not isomorphic to , since the characters differ at .

These account for , so the remaining three irreducible representations have dimensions with , and because and are the only -dimensional ones. Then and , so ; this leaves , or for , and the only way to write one of these as a sum of two squares of integers is . So the remaining dimensions are , and .

14.3.1 Finding the Remaining Irreducibles

The most effective way to find them is to decompose , of dimension , or rather its two pieces and , of dimensions and . By Proposition 13.4.3(vi), which come from the fact that if has eigenvalues on , then it has eigenvalues , , on and , , on . To use them we need . Squaring the representatives: ; and are -cycles; is a double transposition; and is a -cycle. This gives:

First, is irreducible: It is the -dimensional irreducible representation. Next, so is a direct sum of three pairwise non-isomorphic irreducible representations. Two of them are and : So , where is irreducible of dimension , with character Finally, is irreducible with character , and because the characters differ at . This gives all seven irreducible representations.

14.3.2 The Complete Character Table

As checks, , and column orthogonality holds; for instance, the column of gives .