The Lumenary · First-year graduate

Representation theory of finite groups

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You will learn to

Define representations, intertwiners and irreducibility, and identify representations of $G$ with modules over the group algebra $k[G]$
Prove Maschke's theorem and Schur's lemma, and explain why complete reducibility fails when the characteristic divides $|G|$
Derive the orthogonality relations for characters, and deduce that the number of irreducible complex representations equals the number of conjugacy classes and that the squares of their dimensions sum to $|G|$
Decompose representations with characters, and construct the character tables of $S_3$ , $S_4$ , $D_4$ and $Q_8$
Compute induced representations and prove Frobenius reciprocity; show that $D_4$ and $Q_8$ have the same character table

1.
The alternating group $A_4$ (order $12$ ) has $4$ conjugacy classes and exactly $3$ one-dimensional complex representations. What is the degree of its remaining irreducible representation?
2.
Which statements are true for complex representations of a finite group $G$ ? (Choose all that apply.)
- A.If $G$ is abelian, every irreducible representation is one-dimensional
- B.Characters are constant on conjugacy classes
- C. $\chi_V(1) = \dim V$
- D.If $\chi_V(g) = 0$ then $\rho(g) = 0$
- E.Over every field, every representation of $G$ is completely reducible
3.
A representation of $S_3$ has character $\chi = (4, 0, 1)$ on the classes of the identity, the transpositions and the $3$ -cycles (sizes $1, 3, 2$ ). Decompose it into irreducibles ( $\mathbf{1}$ trivial, $\varepsilon$ sign, $V$ two-dimensional).
- A. $\mathbf{1} \oplus \varepsilon \oplus V$
- B. $V \oplus V$
- C. $\mathbf{1} \oplus \mathbf{1} \oplus V$
- D. $\mathbf{1} \oplus \mathbf{1} \oplus \varepsilon \oplus \varepsilon$
4.
$S_4$ permutes the $6$ two-element subsets of $\{1, 2, 3, 4\}$ . The character of this permutation representation on the classes $1, (12)(34), (12), (123), (1234)$ (sizes $1, 3, 6, 8, 6$ ) is $(6, 2, 2, 0, 0)$ . Compute $\langle \chi, \chi \rangle$ , the number of irreducible constituents counted with squared multiplicity.
5.
In the proof of Maschke's theorem we replace a projection $\pi : V \to W$ by $\pi_0 = \frac{1}{|G|}\sum_g g\pi g^{-1}$ . Which step uses that the characteristic of $k$ does not divide $|G|$ ?
- A.Dividing by $|G|$ , so that $\pi_0$ is still the identity on $W$ (a projection onto $W$ )
- B.Showing that $\pi_0$ commutes with the action of $G$
- C.Choosing the linear projection $\pi$ in the first place
- D.Showing that $\pi_0$ maps $V$ into $W$
6.
Which of these representations are not completely reducible? (Choose all that apply.)
- A.The group of order $2$ acting on $\mathbb{F}_2^2$ by $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$
- B.The group of order $3$ acting on $\mathbb{F}_3^2$ by $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$
- C. $\mathbb{Z}$ acting on $\mathbb{C}^2$ by $n \mapsto \left(\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix}\right)$
- D.The group of order $2$ acting on $\mathbb{F}_3^2$ by swapping coordinates
- E. $C_3$ acting on $\mathbb{R}^2$ by rotations through $120^\circ$
- F. $S_3$ permuting the coordinates of $\mathbb{C}^3$
7.
Let $\omega$ be a non-trivial one-dimensional character of $A_3 = \{1, (123), (132)\} \le S_3$ . What is $\operatorname{Ind}_{A_3}^{S_3} \omega$ ?
- A.The irreducible two-dimensional representation $V$
- B. $\mathbf{1} \oplus \varepsilon$
- C.A three-dimensional representation
- D. $\omega \oplus \bar\omega$
8.
Which Lean statement says that the shear representation over $\mathbb{F}_2$ is not completely reducible, as in maschke_fails? (Here L abbreviates Submodule.span (ZMod 2) {((1 : ZMod 2), (0 : ZMod 2))}.)
- A.¬ ∃ q : Submodule (ZMod 2) (ZMod 2 × ZMod 2), IsCompl L q ∧ ∀ x ∈ q, shear x ∈ q
- B.¬ ∃ q : Submodule (ZMod 2) (ZMod 2 × ZMod 2), IsCompl L q
- C.∃ q : Submodule (ZMod 2) (ZMod 2 × ZMod 2), IsCompl L q ∧ ∀ x ∈ q, shear x ∈ q
- D.shear ∘ₗ shear ≠ LinearMap.id
9.
Write multiplicities(chi, table, sizes): given a real character chi (values on the conjugacy classes), the character table (a list of rows of real irreducible characters, same class order) and the class sizes, return the list of multiplicities $m_i = \frac{1}{|G|}\sum_C |C|\,\chi(C)\,\chi_i(C)$ as integers, where $|G| = \sum_C |C|$ .
Computational — Practice

Answer key — Representation theory of finite groups

1.
There are $4$ irreducibles (one per class); with degrees $1, 1, 1, d$ we need $3 + d^2 = 12$ , so $d = 3$ . Geometrically: $A_4$ is the rotation group of the regular tetrahedron, acting irreducibly on $\mathbb{R}^3$ .
2.
True: irreducibles of abelian groups are one-dimensional; characters are class functions; $\chi(1) = \dim V$ . False: $\rho(g)$ is always invertible, so a zero character value does not make it zero (a reflection of $\mathbb{R}^2$ has trace $0$ ); complete reducibility needs $\operatorname{char} k \nmid |G|$ .
3.
$m_{\mathbf{1}} = \frac{1}{6}(4 + 3 \cdot 0 + 2 \cdot 1) = 1$ , $m_\varepsilon = \frac{1}{6}(4 - 0 + 2) = 1$ , $m_V = \frac{1}{6}(8 + 0 - 2) = 1$ , so $\chi = \chi_{\mathbf{1}} + \chi_\varepsilon + \chi_V$ ; check the degree $1 + 1 + 2 = 4$ . This is $V \otimes V$ .
4.
$\langle \chi, \chi \rangle = \frac{1}{24}(1 \cdot 36 + 3 \cdot 4 + 6 \cdot 4 + 0 + 0) = 3$ , so $\chi$ is a sum of three distinct irreducibles, each once. Inner products with the table give $\chi = \mathbf{1} + (2, 2, 0, -1, 0) + (3, -1, 1, 0, -1)$ , and the degrees add up: $1 + 2 + 3 = 6$ .
5.
$\sum_g g\pi g^{-1}$ maps into $W$ and commutes with $G$ in any characteristic, but on $W$ it is multiplication by $|G|$ . To get a projection onto $W$ one divides by $|G|$ , which requires $\operatorname{char} k \nmid |G|$ . In characteristic $2$ the experiment shows the sum is then not a projection, and indeed the shear representation has no invariant complement.
6.
Not completely reducible: the shear for $p = 2$ and $p = 3$ (characteristic divides $|G| = p$ ) and the infinite group $\mathbb{Z}$ acting unipotently. Completely reducible: the swap over $\mathbb{F}_3$ , the rotation representation of $C_3$ over $\mathbb{R}$ (irreducible), and the permutation representation of $S_3$ over $\mathbb{C}$ — all covered by Maschke's theorem.
7.
$\operatorname{Ind}\omega$ has degree $2$ . Restricted to $A_3$ , $\mathbf{1}$ and $\varepsilon$ are trivial, so $\langle \operatorname{Ind}\omega, \mathbf{1} \rangle = \langle \operatorname{Ind}\omega, \varepsilon \rangle = \langle \omega, \mathbf{1} \rangle_{A_3} = 0$ , while $\operatorname{Res}\chi_V = (2, -1, -1) = \omega + \bar\omega$ gives $\langle \operatorname{Ind}\omega, \chi_V \rangle = 1$ . So $\operatorname{Ind}\omega \cong V$ . Using the formula, its character is $(2, 0, \omega(123) + \omega(132)) = (2, 0, -1)$ .
8.
The correct statement negates the existence of a complement q of L that is invariant under shear. Linear complements exist (so the version without invariance is false), and shear ∘ₗ shear = LinearMap.id holds. The Lean proof: a complement contains some $b = (c, 1)$ ; invariance would put $\mathrm{shear}(b) - b = (1, 0)$ in $q \cap L = 0$ .
9.
```
def multiplicities(chi, table, sizes):
    order = sum(sizes)
    return [sum(s * a * b for s, a, b in zip(sizes, chi, row)) // order
            for row in table]
```
The regular character $(6, 0, 0)$ of $S_3$ contains each irreducible as often as its degree, $(1, 1, 2)$ , which is Corollary 4(3); the permutation character $(4, 0, 2, 1, 0)$ of $S_4$ on four points is trivial plus the standard representation. For complex-valued characters one would use $\overline{\chi_i(C)}$ in the sum.