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The character table of $S_{3}$

Example

The character table of $S_{3}$ , with columns indexed by the classes of $1$ , $(12)$ , and $(123)$ (sizes $1$ , $3$ , $2$ ), is

$\begin{array}{cccc} 1 & (12) & (123) \\ 1 & 1 & 1 & 1 \\ sgn & 1 & - 1 & 1 \\ χ_{s t d} & 2 & 0 & - 1 \end{array}$

Both orthogonality relations hold for this table.

Facts & Assumptions

Given: The symmetric group $S_{3}$ with its three irreducible characters $1$ , $sgn$ , $χ_{s t d}$ .

[F1]

$S_{3}$ has exactly three irreducible characters of degrees $1$ , $1$ , and $2$ , and its classes have sizes $1$ , $3$ , $2$ ( $S_{3}$ has three irreducible complex characters of degrees $1$ , $1$ , and $2$ ).

[F2]

The first orthogonality relation: the rows are orthonormal (The first orthogonality relation for irreducible complex characters).

[F3]

The second orthogonality relation: distinct columns are orthogonal and the squared norm of a column is the centralizer size (The second orthogonality relation for irreducible complex characters).

[A1]

$sgn (σ) = 1$ for even $σ$ and $- 1$ for odd $σ$ , so on the three class representatives it is $1$ , $- 1$ , $1$ .

[A2]

The standard character is $fix (σ) - 1$ , with values $2$ , $0$ , $- 1$ on the three representatives.

Verification

technique · direct

1.1

By [F1] there are three irreducible characters and three classes; [A1] and [A2] supply the second and third rows, and the trivial character is the constant $1$ . Hence the displayed table is the character table.

F1A1A2given

2.1

Row orthogonality: the rows $1$ and $sgn$ have inner product $\frac{1}{6} (1 - 3 + 2) = 0$ , the row $1$ with $χ_{s t d}$ gives $\frac{1}{6} (2 + 0 - 2) = 0$ , $sgn$ with $χ_{s t d}$ gives $\frac{1}{6} (2 - 0 - 2) = 0$ , and each row has self-inner-product $1$ ; this matches [F2].

F2step 1.1algebra

3.1

Column orthogonality: the first column has squared norm $1 + 1 + 4 = 6 = ∣ C_{S_{3}} (1) ∣$ ; the second has $1 + 1 + 0 = 2 = ∣ C_{S_{3}} ((12)) ∣$ ; the third has $1 + 1 + 1 = 3 = ∣ C_{S_{3}} ((123)) ∣$ ; and each pair of distinct columns has inner product $1 - 1 + 0 = 0$ , $1 + 1 - 2 = 0$ , $1 - 1 + 0 = 0$ , matching [F3].

F3step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Example 3.1.2 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Section 3.5 (standard reference, not scraped)

The character table of S3S3​