Appendix D — Homework 3

Exercise D.1 Let \(G\) be a group and let \(\rho: G \to \operatorname{GL}(V)\) and \(\sigma: G \to \operatorname{GL}(W)\) be representations. If \(\varphi: V \to W\) is an intertwiner, show that \[ \operatorname{im}\varphi \leq_G W. \]

Exercise D.2 A division algebra \(D\) over a field \(F\) is a (not necessarily commutative) \(F\)-algebra such that every non-zero element has a multiplicative inverse. Note that any such algebra contains \(F\) as a sub-algebra, by identifing \[ \begin{aligned} F & \hookrightarrow D \\ \alpha & \mapsto \alpha 1_D, \end{aligned} \] where \(1_D \in D\) is the multiplicative identity.

Let \(D\) be a finite-dimensional division algebra over \(\mathbb{C}\). Show that \(D = \mathbb{C}\).

Hint: If \(D\) is an \(n\)-dimensional \(\mathbb{C}\)-algebra and \(x \in D\), the set \(\{1_D, x, \dots, x^n \}\) must be linearly dependent. You may use the fact that division algebras cannot contain zero divisors.

Exercise D.3 Let \(G\) be a finite group with \(V\) an irreducible representation over \(\mathbb{C}\). Consider \[ \operatorname{End}_G(V) \coloneqq \{ L \in \operatorname{End}(V) \mid L(g \cdot x) = g \cdot Lx \text{ for all } g \in G, x \in V \}, \] the algebra of \(G\)-linear maps on \(V\).1 Assuming the first part of Schur’s Lemma (Lemma 5.1), show that \(\operatorname{End}_G(V)\) is a division algebra over \(\mathbb{C}\) and conclude the second part of Schur’s Lemma by Exercise D.2.

Exercise D.4 Let \(G\) be a finite group with \(V\) a finite-dimensional complex \(G\)-representation. Suppose that \(\langle \cdot, \cdot \rangle_G\) is a \(G\)-invariant inner product, as guaranteed by Weyl’s unitary trick (Lemma 4.1), and suppose \(\langle \cdot, \cdot \rangle_0\) is another \(G\)-invariant inner product. We will show that \[ \langle \cdot, \cdot \rangle_0 = \lambda \langle \cdot, \cdot \rangle_G \] for some \(\lambda \in \mathbb{R}^+\) whenever \(V\) is irreducible, i.e., the \(G\)-invariant inner product is unique up to scaling.

  1. For a fixed \(x \in V\), the map \(y \mapsto \langle x, y \rangle_G\) is a linear functional \(V \to \mathbb{C}\). By applying Riesz representation (Theorem 2.11) to the inner product space \((V, \langle \cdot, \cdot \rangle_0)\), we can represent this functional by some unique \(x' \in V\): \[ \langle x, y \rangle_G = \langle x', y \rangle_0 \] for all \(y \in V\) (note the different inner products above). This process defines a map: \[ \begin{aligned} \phi: V & \to V \\ x & \mapsto x'. \end{aligned} \] Show that \(\phi\) is a linear isomorphism.

  2. Show that \(\phi\) is an intertwiner, i.e., that \(\phi(g \cdot x) = g \cdot \phi(x)\) for all \(g \in G\) and \(x \in V\).

  3. Suppose \(V\) is irreducible. Use Schur’s Lemma to show there is a \(\lambda \in \mathbb{R}^+\) such that \(\langle \cdot, \cdot \rangle_0 = \lambda \langle \cdot, \cdot \rangle_G\).

Exercise D.5 Let \(\rho: G \to \operatorname{GL}(V)\) be a representation. Consider the Reynolds operator, \[ \varphi \coloneqq \frac{1}{|G|} \sum_{h \in G} \rho(h), \] given by averaging over the linear maps in \(\operatorname{im}(G)\).

  1. Show that \(\varphi\) is a projection onto the fixed subspace \(V^G\).
  2. Show that \(\varphi\) is an orthogonal projection with respect to \(\langle \cdot, \cdot \rangle_G\).

Exercise D.6 Let \(G\) be a finite group and consider the regular representation \(\rho: G \to \operatorname{GL}(\mathbb{C}[G])\). Compute \(\operatorname{Tr}\rho(g)\) for every \(g \in G\) and use Exercise D.5 to compute the dimension of the fixed subspace of \(\mathbb{C}[G]\).

Hint: Appeal to Exercise A.6 and Exercise B.1.

Exercise D.7 In order to develop a calculus of group representations, one of the core results of this course, we want to attach invariants to representations. These are objects (numbers, functions, etc.) that, in particular, do not depend on which basis we choose to represent the transformations \(\rho(g)\). The primary invariant we will use is the trace of a representation, though others exist!

Let \(G\) be a group and consider all its finite-dimensional representations \(\rho: G \to \operatorname{GL}(V)\), which we want to understand and classify. Instead of the trace, explain why determinant maps \(G \to F\) given by \(g \mapsto \det \rho(g)\) might not be a fruitful invariant to study.

Remark: You can do this, for example, by enumerating families of non-isomorphic representations of a group \(G\) whose determinants agree for all \(g \in G\), or by appealing to Exercise C.1.

  1. Recall that \(\operatorname{End}(V) \coloneqq \operatorname{Hom}(V,V)\) is the space of linear operators on \(V\)—in particular, \(\dim \operatorname{End}(V) = (\dim V)^2\).↩︎