14.3 Character Table of

has 7 conjugacy classes, determined by cycle type:

So there are 7 irreducible representations. Since , we need to find dimensions summing in squares to 120.

We start from known representations:

• : trivial, dim = 1.
• : sign (alternating), dim = 1.
• : standard representation. The permutation representation decomposes as where , so . Its character is .
• : another 4-dimensional irreducible with .

These account for , leaving . Since we need three more squares summing to 86, the only possibility is . So the remaining dimensions are 5, 5, 6.

14.3.1 Finding the Remaining Irreducibles via Tensor Products

The most effective method is to analyze (dim = 16), or rather its two pieces: (dim = 10) and (dim = 6).

If has eigenvalues , then has eigenvalues for , and has eigenvalues for . The resulting character formulas are:

We compute these for the standard representation of , whose character is . To apply the formulas, we need . Squaring cycle types: , , (a 3-cycle), , (double transposition), (a 5-cycle).

We check irreducibility of : So is irreducible.

For : So splits into 3 irreducible summands. Computing inner products with known characters: while and . So where is a new irreducible of dimension . Its character is

Finally, gives another 5-dimensional irreducible with .

14.3.2 The Complete Character Table

Check: . The column sums of weighted by class size also verify the orthogonality relations.