What is representation?

Example 5.2

(a) Degree $1$: morphisms $G \to \C^\times$. (b) The regular representation: $V = \C^G$ with basis $(e_h)_{h \in G}$, $\rho(g)e_h = e_{gh}$; degree $\abs G$. (c) A permutation action of $G$ on a finite set $X$ gives the permutation representation on $\C^X$: $\rho(g)e_x = e_{g\cdot x}$. (d) $S_n$ acts on $\C^n$ by permuting coordinates; the hyperplane $\{\sum x_i = 0\}$ is invariant: the standard representation, of degree $n - 1$.

Example 5.15 (The table of $S_3$)

Classes: $e$ (size 1), transpositions (3), $3$-cycles (2); so $r = 3$ irreducibles, of degrees $n_i$ with $\sum n_i^2 = 6$: $1, 1, 2$. Degree $1$: trivial $\mathbf 1$ and signature $\varepsilon$. The last row follows from column orthogonality (or from $\chi_{\mathrm{std}} = \chi_{\mathrm{perm}} - \mathbf 1$):

Check: $\langle\chi_{\mathrm{std}},\chi_{\mathrm{std}}\rangle = \frac{1}{6}(4 + 0 + 2) = 1$: irreducible.

Example 5.16 (The table of $S_4$)

Classes: $e$ [1], transpositions [6], double transpositions [3], $3$-cycles [8], $4$-cycles [6]: five irreducibles, $\sum n_i^2 = 24$ with two degree-$1$’s ($\mathbf 1, \varepsilon$; $[S_4 : D(S_4)] = [S_4 : A_4] = 2$): degrees $1, 1, 2, 3, 3$. The degree-$2$ lifts from $S_4/V \cong S_3$ (Proposition 5.13(b), $V$ the Klein group); degree $3$: the standard representation and its twist by $\varepsilon$:

($\chi_{\mathrm{std}}(g) = \operatorname{fix}(g) - 1$; $\chi_2$ evaluates the $S_3$-table on the image of each class mod $V$.) All row and column orthogonality checks pass — running two of them is Exercise 5.3’s warmup.