With the results of Chapter 13 we can compute the character table of a given group without first constructing all of its representations. We use the notation of that chapter: are the irreducible representations of , their characters, and the inner product on class functions (Definition 13.5.3). The tools are these.
- The number of irreducible representations equals the number of conjugacy classes (Theorem 13.6.2), and the squares of their dimensions add up to (Corollary 13.5.11).
- The irreducible characters are orthonormal (Theorem 13.5.4), and the columns of the table are orthogonal (Proposition 13.6.5).
- A representation is irreducible exactly when (Corollary 13.5.7), and is the multiplicity of in (Corollary 13.5.6).
- New representations come from old ones: permutation representations, twists by -dimensional representations (Proposition 13.5.8), tensor products, symmetric and exterior squares, whose characters are given by Proposition 13.4.3.
We work out the character tables of , , and in this way; the table of is Example 13.4.5. The chapter ends with the representation ring, which packages the operations and into a ring.
For a symmetric group, , and denote its trivial, alternating and standard representations (Example 13.1.4), and , because is the permutation representation, whose character counts fixed points. In the tables, each column is labelled by a representative of a conjugacy class, with the size of the class below it. When we write a class function as a vector, its entries are its values on the classes in the order of the table.
Two general facts about -dimensional representations will be used repeatedly. First, if is -dimensional and is irreducible, then is irreducible with character (Proposition 13.5.8). Second, () has exactly two -dimensional representations, and . Indeed, a homomorphism takes the same value on all transpositions, since they are conjugate (Theorem 3.4.6) and is abelian; and . As the transpositions generate (Proposition 3.4.9), is the constant function if , and if .
By Theorem 3.4.6, the conjugacy classes of are given by cycle type: the identity, the six transpositions, the three double transpositions such as , the eight -cycles and the six -cycles. So has five irreducible representations, and the squares of their dimensions add up to .
We know three of them. The rows of and are the constant function and , and in the order of the table below. Since is irreducible. The twist is irreducible with character , and because the characters differ at . So far , so the fifth irreducible representation has dimension .
To find , observe that is also a -dimensional irreducible representation, and is the only one; so and . Hence vanishes on the odd classes, those of and . Write and . Orthogonality to and to gives so and . The table is complete:
| | | | | |
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| size | 1 | 6 | 3 | 8 | 6 |
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| 1 | 1 | 1 | 1 | 1 |
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| 1 | | 1 | 1 | |
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| 2 | 0 | 2 | | 0 |
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| 3 | 1 | | 0 | |
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| 3 | | | 0 | 1 |
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As a check, .
14.1.1 Identifying
The value tells us a great deal. The operator by which acts on has order dividing , so it is diagonalizable with eigenvalues (Lemma 13.1.5), and two such eigenvalues add up to only if both are . So acts on as the identity, and so do the other two double transpositions, which have the same character value. Hence the subgroup lies in the kernel of . It is normal, being the union of the conjugacy classes of and of (Proposition 3.2.13), so this homomorphism factors through (Theorem 3.3.2).
To identify , let act on the three ways of splitting into two pairs, This gives a homomorphism . Each double transposition preserves all three splittings, so . The transposition fixes and swaps and , and permutes , so the image of contains elements of orders and , and is surjective. Then , so , and by the First Isomorphism Theorem (Theorem 3.3.3). Under , transpositions and -cycles go to transpositions (for instance, fixes and swaps and ), and -cycles go to -cycles.
Now let be the standard representation of , regarded as a representation of in which acts as does. Its character is , where is the standard representation of ; by Example 13.4.5 this is , which is . So (Corollary 13.5.6): is the standard representation of , pulled back to along .
14.1.2 A Second Construction of
We can also find inside . Its character is , and So contains , of dimension , and the remaining summand has dimension and contains none of ; it must be . Thus and , which again gives . (Consistently, , the value for four non-isomorphic summands.)