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Let \(\varphi:D_{4} \to GL_{2}(\mathbb{C})\) be the given by \[ \varphi(r^k) = \begin{bmatrix} i^k & 0 \\ 0 & (-i)^k \end{bmatrix}, \varphi(sr^k) = \begin{bmatrix} 0 & (-i)^k \\ i^k & 0 \end{bmatrix} \] where \(r\) is the rotation counterclockwise by \(\pi/2\) and \(s\) is the reflection over the \(x\)-axis. Prove that \(\varphi\) is irreducible

Since \(\varphi(r)\) is a diagonal matrix, the eigenvalues are its nonzero entries and the eigenvectors are the elements of the standard basis. Since \(\varphi_{s}\) rotates vectors by \(\pi/2\), neither standard basis vector is an eigenvector of \(varphi_{s}\). Since \(D_{4}\) is generated by \(r,s\) and they do not share a common eigenvector, Proposition 3.1.19 gives \(\varphi\) is irreducible.

Let \(\varphi, \psi:G \to \mathbb{C}^*\) be one-dimensional representations show that \(\varphi\) is equivalent to \(\psi\) if and only if \(\varphi = \psi\).

Any isomorphism \(T: \mathbb{C} \to \mathbb{C}\) is simply multiplication by an element in \(\mathbb{C}^*\). Further, \(\varphi_{g}, \psi_{g} \in \mathbb{C}^*\) so \(\psi_{g}z\) is simply multiplying \(z\) be some complex number. Hence, for all \(g \in G\), \[ T\psi_{g}zT^{-1} = \phi_{g}z ~\forall z \in \mathbb{C} \iff \psi_{g} = \phi_{g}. \]
Let \(\varphi:G \to \mathbb{C}^*\) be a representation. Suppose that \(g \in G\) has order \(n\).
  1. Show that \(\varphi(g)\) is an \(n\)th-root of unity.
  2. Construct \(n\) inequivalent one-dimensional representations \(\mathbb{Z}/n\mathbb{Z} \to \mathbb{C}^*\).
  3. Explain why your representations are the only possible one-dimensional representations \(\mathbb{Z}/n\mathbb{Z} \to \mathbb{C}^*\).
  1. Homomorphisms preserve group structure so \((\varphi_{g})^n = \varphi_{g^n} = \varphi_{e} = 1\).
  2. From Exercise 3.3, two representations \(\mathbb{Z}/n\mathbb{Z} \to \mathbb{C}^*\) are equivalent if and only if they are equal. \(\varphi_{[m]} = e^{i \pi m/n}\).
  3. Only possible becuase must be roots of unity?
Let \(V\) be a vector space and \(P \in \text{End}(V)\) such that \(P^2 = P\).
  1. Show that \(V\) is the internal direct sum of \(\ker(P)\) and \(\img(P)\)
  2. Show that the trace of \(P\) is equal to the rank of \(P\) (i.e., to \(\text{dim}(\img(P))\).
There is a hint for (B)
Let \(\varphi:G \to GL(V)\) be a representation of a finite group \(G\). Define the fixed subspace \[ V^G = \{ v \in V \mid \varphi_{g}v = v, ~\forall g \in G \}. \]
  1. Show that \(V^G\) is a \(G\)-invariant subspace.
  2. Show that
\[ \frac{1}{|G|} \sum_{h \in G}\varphi_{h}v \in V^G \] for all