Appendix G — Homework 6

Exercise G.1 Let \(G\) and \(H\) be finite groups. Prove that the irreducible representations of \(G \times H\) are all of the form \(V \otimes W\), where \(V\) and \(W\) are irreducible representations of \(G\) and \(H\), respectively.

Remark: You need to show two things. Firstly, if \(\rho:G \to \operatorname{GL}(V)\) and \(\sigma: H \to \operatorname{GL}(W)\) are irreducible representations, you should show that the map \(G \times H \to \operatorname{GL}(V \otimes W)\) given by \((g,h) \mapsto \rho(g) \otimes \sigma(h)\) is an irreducible representation. Secondly, you should explain why any irreducible representation of \(G \times H\) is of this form. It might be helpful to prove some other lemmas!

Exercise G.2 Let \(G\) be a finite group.

  1. By Noether’s theorems, for any \(N \unlhd G\), the representations of \(G\) with kernels containing \(N\) are in one-to-one correspondence with representations of \(G/N\). One direction is simply composing with the projection map \(\pi: G \twoheadrightarrow G/N\); the other is associating \(\tilde{\rho}: G \to \operatorname{GL}(V)\) to the (well-defined) homomorphism \[ \begin{aligned} \rho: G/N &\to \operatorname{GL}(V) \\ gN &\mapsto \tilde{\rho}(g), \end{aligned} \] where we call \(\tilde{\rho}\) the lift of \(\rho\). Show that \(\rho\) is irreducible if and only if \(\tilde{\rho}\) is irreducible.

  2. Prove that every normal subgroup of G can be written as \[ N = \bigcap_{V \in S} \ker \rho_V, \] where \(S\) is some subset of the (isomorphism classes of) irreducible \(G\)-representations. In this sense, the character table of \(G\) captures all normal subgroups.

    Hint: From column orthogonality, you can argue that the intersection over all irreducible \(G\)-representations is \(\{e\}\).

Exercise G.3 Below is the character table of a group \(G\).

\[ \begin{array}{c|rrrrrrr} \text{size} & 1 & 1 & 4 & 2 & 4 & 2 & 2 \\ \text{class} & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 \\ \hline V_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ V_2 & 1 & 1 & 1 & 1 & -1 & -1 & -1 \\ V_3 & 1 & 1 & -1 & 1 & 1 & -1 & -1 \\ V_4 & 1 & 1 & -1 & 1 & -1 & 1 & 1 \\ V_5 & 2 & 2 & 0 & -2 & 0 & 0 & 0 \\ V_6 & 2 & -2 & 0 & 0 & 0 & -\sqrt{-2} & \sqrt{-2} \\ V_7 & 2 & -2 & 0 & 0 & 0 & \sqrt{-2} & -\sqrt{-2} \\ \end{array} \]

Determine the normal subgroup lattice of \(G\).

Exercise G.4 An element \(g \in G\) is called real if \(g\) is conjugate to its inverse, \(g^{-1}\). An irreducible character \(\chi\) is called real-valued if \(\chi(g) \in \mathbb{R}\) for all \(g \in G\). Here we characterize the real-valued irreducible representations of \(G\).

  1. Show that \(g \mapsto g^{-1}\) induces a permutation of conjugacy classes of \(G\). A conjugacy class fixed by this action is called real.

  2. If \(V\) is irreducible, show that \(V^*\) is irreducible. Conclude that \(\chi \mapsto \overline{\chi}\) induces a permutation of irreducible characters.

  3. Write \(M\) for the character table of \(G\) regarded as a matrix and enumerate representatives of the conjugacy classes, \(g_1, \dots, g_r\). Let \(P, Q\) be the \(r \times r\) permutation matrices corresponding to the action of complex conjugation and taking inverses, respectively, so that:

    • The \(i\)th row of \(PM\) corresponds to the character \(\overline{\chi_i}\)
    • The \(i\)th column of \(MQ\) corresponds to the conjugacy class of \((g_i)^{-1}\).

    Prove that \(PM = MQ\) and conclude that \(\operatorname{Tr}(P) = \operatorname{Tr}(Q)\).

    Hint: Is \(M\) invertible?

  4. Show that the number of real-valued irreducible representations of \(G\) is equal to the number of real conjugacy classes. Conclude that if \(G\) has \(r-k\) real conjugacy classes, then \(G\) has exactly \(k\) irreducible characters that are not real-valued, partitioned into \(k/2\) complex pairs \(\chi\) and \(\overline{\chi}\).

Exercise G.5 We can think of the group algebra \(\mathbb{C}[G]\) as the space1 \(\mathscr{L}^{2}({G})\) of functions \(G \to \mathbb{C}\), where \[ \sum_{h \in G} \alpha_h e_h\ \xleftrightarrow{\text{corresponds to}}\ \bigl(g \mapsto \alpha_g\bigr). \] In other words, the basis element \(e_h \in \mathbb{C}[G]\) can be thought of as the indicator function \[ \mathbb{1}_{\{h\}}(g) = \left\{ \begin{array}{ll} 1 & \text{if } h = g \\ 0 & \text{otherwise.} \end{array} \right. \]

  1. Under this identification, show that the center2 of \(\mathbb{C}[G]\) is exactly \(\mathscr{C}({G})\).

    Hint: If \(g_1, \dots, g_r\) are representatives of the conjugacy classes of \(G\), you can argue that elements of the form \[ c_i \coloneqq \sum_{{h \in \operatorname{cl}_{}({g_i})}} e_h \quad \text{ and } \quad \mathbb{1}_{\operatorname{cl}_{}({g_i})} \] are a basis for \(\mathbf{Z}(\mathbb{C}[G])\) and \(\mathscr{C}({G})\), respectively.

  2. Given \(f: G \to \mathbb{C}\) and representation \(\rho: G \to \operatorname{GL}(V)\), define the operator \[ \hat{f} = \sum_{h \in G} f(h) \rho(h): V \to V. \] Show that, whenever \(f\) is a class function, \(\hat{f}\) is an intertwiner.

  1. We write \(\mathscr{L}^{2}({G})\), which technically denotes the inner product space of Definition 7.2 studied throughout the course, because this set of functions enjoys many structures!↩︎

  2. Much like the definition for groups, the center of a ring \(R\) is the set \(\mathbf{Z}(R) \coloneqq \{ r \in R: xr = rx \text{ for all } x \in R \}\).↩︎