Week 2 Exercises

Some questions we left open last time:

  1. Let \rho: \mathbb{Z}/n \to \mathbb{C}^* be given by 1 \mapsto \zeta for some n-th root of unity \zeta. Show that \rho^* is 1 \mapsto \bar{\zeta}. Conclude that \rho, \rho^* are not isomorphic.
  2. Let A be an n \times n matrix of finite order (A^m = I for some m \in \mathbb{Z}). Show that tr(A^{-1}) = \overline{tr(A)}, without using eigenvalues.
  3. In the proof of Prop. 1 of Serre (see Section 2.1), he states that “… \rho_s can be defined to be a unitary matrix”. Show that every representation of a finite group is unitary. (As part of this exercise, give an appropriate definition of a unitary representation.)
  4. Prove that the character of V_1 \otimes V_2 is given by \chi_1 \cdot \chi_2. If \rho is a representation, prove that the character of a dual representation is \chi_{\rho^*} = \overline{\chi_\rho}. Find the characters of \textrm{Sym}^2(\rho) and \rho \wedge \rho = \textrm{Alt}^2(\rho).
  5. If f: (\rho_1, V_1) \to (\rho_2, V_2) is a map of representations, show that the kernel and image of f are subrepresentations of (\rho_1, V_1), (\rho_2, V_2) respectively.
  6. Midterm question: Give your own proof of Serre Thm. 3 (section 2.3), that characters of irreducible representations form an orthonormal system.
  7. Prove Prop. 6 of Serre (section 2.5): If (\rho, V) is an irreducible representation on G of degree n and character \chi, and f is a class function on G, then \rho_f := \sum_{t \in G} f(t)\rho(t) is a homothety of ratio \frac{1}{n} \sum_{t \in G} f(t) \chi(t).
  8. Prove that the number of degree 1 representations on a finite group G is |G^{ab}|.

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