Work any 3 of the following 4 problems.
In these exercises, \(G\) always denotes a finite group. Unless indicated otherwise, all vector spaces are assumed to be finite dimensional over the field \(F = \mathbb{C}\). The representation space \(V\) of a representation of \(G\) is always assumed to be finite dimensional over \(\mathbb{C}\).
Let \(\phi:G \to F^\times\) be a group homomorphism; since \(F^\times = \operatorname{GL}_1(F)\), we can think of \(\phi\) as a 1-dimensional representation \((\phi,F)\) of \(G\).
If \(V\) is any representation of \(G\), we can form a new representation \(\phi \otimes V\). The underlying vector space for this representation is just \(V\), and the “new” action of an element \(g \in G\) on a vector \(v\) is given by the rule \[g \star v = \phi(g) gv.\]
Prove that if \(V\) is irreducible, then \(\phi \otimes V\) is also irreducible.
Prove that if \(\chi\) denotes the character of \(V\), then the character of \(\phi \otimes V\) is given by \(\phi \cdot \chi\); in other words, the trace of the action of \(g \in G\) on \(\phi \otimes V\) is given by \[\chi_{\phi \otimes V}(g) = \operatorname{tr}( v\mapsto g \star v) = \phi(g) \chi(g).\]
Recall that in class we saw that \(S_3\) has an irreducible representation \(V_2\) of dimension 2 whose character \(\psi_2\) is given by
\[\begin{array}{l|lll} g & 1 & (12) & (123) \\ \hline \psi_2 & 2 & 0 & -1 \end{array}\]
Observe that \(\operatorname{sgn} \psi = \psi\) and conclude that \(V_2 \simeq \operatorname{sgn} \otimes V_2\), where \(\operatorname{sgn}:S_n \to \{\pm 1\} \subset \mathbb{C}^\times\) is the sign homomorphism.
On the other hand, \(S_4\) has an irreducible representation \(V_3\) of dimension 3 whose character \(\psi_3\) is given by
\[\begin{array}{l|lllll} g & 1 & (12) & (123) & (1234) & (12)(34) \\ \hline \psi_3 & 3 & 1 & 0 & -1 & -1 \end{array}\]
(I’m not asking you to confirm that \(\psi_3\) is irreducible, though it would be straightforward to check that \(\langle \psi_3,\psi_3 \rangle = 1\)).
Prove that \(V_3 \not \simeq \operatorname{sgn} \otimes V_3\) as \(S_4\)-representations.
(In particular, \(S_4\) has at least two irreducible representations of dimension 3.)
Let \(V\) be a representation of \(G\).
For an irreducible representation \(L\), consider the set \[\mathcal{S}=\{ S \subseteq V \mid S \simeq L\}\] of all invariant subspaces that are isomorphic to \(L\) as \(G\)-representations.
Put \[V_{(L)} = \sum_{S \in \mathcal{S}} S.\]
Prove that \(V_{(L)}\) is an invariant subspace, and show that \(V_{(L)}\) is isomorphic to a direct sum \[V_{(L)} \simeq L \oplus \cdots \oplus L\] as \(G\)-representations.
Prove that the quotient representation \(V/V_{(L)}\) has no invariant subspaces isomorphic to \(L\) as \(G\)-representations.
If \(L_1,L_2,\cdots,L_m\) is a complete set of non-isomorphic irreducible representations for \(G\), prove that \(V\) is the internal direct sum \[V = \bigoplus_{i=1}^m V_{(L_i)}.\]
Let \(\chi\) be the character of a representation \(V\) of \(G\). For \(g\in G\) prove that \(\overline{\chi(g)} = \chi(g^{-1})\).
Is it true for any arbitrary class function \(f:G \to \mathbb{C}\) that \(\overline{f(g)} = f(g^{-1})\) for every \(g\)? (Give a proof or a counterexample…)
For a prime number \(p\), let \(k=\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\) be the field with \(p\) elements. Let \(V\) be an \(n\)-dimensional vector space over \(\mathbb{F}_p\) for some natural number \(n\), and let \[\langle \cdot,\cdot \rangle: V \times V \to k\] be a non-degenerate bilinear form on \(V\).
(A common example would be to take \(V = \mathbb{F}_{p^n}\) the field of order \(p^n\), and \(\langle \alpha,\beta\rangle = \operatorname{tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(\alpha \beta)\) the trace pairing).
Let us fix a non-trivial group homomorphism \(\psi:k \to \mathbb{C}^\times\) (recall that \(k = \mathbb{Z}/p\mathbb{Z}\) is an additive group, while \(\mathbb{C}^\times\) is multiplicative). Thus \[\psi(\alpha + \beta) = \psi(\alpha)\psi(\beta) \quad \text{for all} \quad \alpha,\beta \in k.\] If you want an explicit choice, set \(\psi(j + p\mathbb{Z}) = \exp(j \cdot 2 \pi i/p) = \exp(2 \pi i /p)^j.\)
For a vector \(v \in V\), consider the mapping \(\Psi_v:V \to \mathbb{C}^\times\) given by the rule \[\Psi_v(w) = \psi( \langle w,v \rangle ).\]
Show that \(\Psi_v\) is a group homomorphism \(V \to \mathbb{C}^\times\).
Show that the assignment \(v \mapsto \Psi_v\) is injective (one-to-one).
(This assignment is a function \(V \to \operatorname{Hom}(V,\mathbb{C}^\times)\). In fact, it is a group homomorphism. Do you see why? How do you make \(\operatorname{Hom}(V,\mathbb{C}^\times)\) into a group?)
Show that any group homomorphism \(\Psi:V \to \mathbb{C}^\times\) has the form \(\Psi = \Psi_v\) for some \(v \in V\).
Conclude that there are exactly \(|V| = q^n\) group homomorphisms \(V \to \mathbb{C}^\times\).