PS01--rep-theory.md

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title: |
 ProblemSet 1 -- Linear algebra and representations
author: George McNinch
date: 2024-01-29

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$F$ denotes an algebraically closed field of characteristic 0. If you like, you can suppose that $F = \mathbf{C}$ is the field of complex numbers.

1. Let $V$ be a finite dimensional vector space over the field $F$. Suppose that $\phi,\psi:V \to V$ are linear maps. Let $\lambda \in F$ be an eigenvalue of $\phi$ and write $W$ for the $\lambda$-eigenspace of $\phi$; i.e. $$W = \{v \in V \mid \phi(v) = \lambda v \}.$$ If $\phi \psi = \psi \phi$ show that $W$ is invariant under $\psi$ -- i.e. show that $\psi(W) \subseteq W$.

2. Let $n \in \mathbf{N}$ be a non-zero natural number, and let $V$ be an $n$ dimensional $F$-vector space with a given basis $e_1,e_2,\cdots,e_n$.

   Consider the linear transformation $T:V \to V$ given by the rule $$Te_i = e_{i+1 \pmod n}.$$ In other words $$Te_i = \left \{ \begin{matrix} e_{i+1} & i < n \\ e_1 & i =n \end{matrix} \right ..$$

   a. Show that $T$ is invertible and that $T^n = \operatorname{id}_V$.

   b. Consider the vector $v_0 = \displaystyle \sum_{i=1}^n e_i$. Show that $v_0$ is a $1$-eigenvector for $T$.

   Let $\zeta \in F$ be a primitive $n$-th root of unity. (e.g. if you assume $F = \mathbf{C}$, you may as well take $\zeta = e^{2\pi i/n}$).

   c. Let $v_1 = \displaystyle \sum_{i=1}^n \zeta^i e_i$. Show that $v_1$ is a $\zeta^{-1}$-eigenvector for $T$.

   d. More generally, let $0 \le j < n$ and let $$v_j = \sum_{i=1}^n \zeta^{ij} e_i.$$ Show that $v_j$ is a $\zeta^{-j}$-eigenvector for $T$.

   e. Conclude that $v_0,v_1,\cdots,v_{n-1}$ is a basis of $V$ consisting of eigenvectors for $T$, so that $T$ is diagonalizable.

   Hint: You need to use the fact that eigenvectors for distinct eigenvalues are linearly independent.

   What is the matrix of $T$ in this basis?

3. Let $G = \mathbb{Z}/3\mathbb{Z}$ be the additive group of order $3$, and let $\zeta$ be a primitive $3$rd root of unity in $F$.

   To define a representation $\rho:G \to \operatorname{GL}_n(F)$, it is enough to find a matrix $M \in \operatorname{GL}_n(F)$ with $M^3 = 1$; in turn, $M$ determines a representation $\rho$ by the rule $\rho(i + 3\mathbb{Z}) = M^i$.

   Consider the representation $\rho_1 : G \to \operatorname{GL}_3(F)$ given by the matrix $$\rho_1(1 + 3\mathbb{Z}) = M_1 = \begin{bmatrix} 1 & 0 & 0\\ 0 & \zeta & 0 \\ 0 & 0 & \zeta^2 \end{bmatrix}$$ and consider the representation $\rho_2:G \to \operatorname{GL}_3(F)$ given by the matrix $$\rho_2(1 + 3\mathbb{Z}) = M_2 = \begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}.$$

   Show that the representations $\rho_1$ and $\rho_2$ are equivalent (alternative terminology: are isomorphic). In other words, find a linear bijection $\Phi:F^3 \to F^3$ with the property that $$\Phi(\rho_2(g)v) = \rho_1(g)\Phi(v)$$ for every $g \in G$ and $v \in F^3$.

   Hint: First find a basis of $F^3$ consisting of eigenvectors for the matrix $M_2$.

4. Let $V$ be a $n$ dimensional $F$-vector space for $n \in \mathbb{N}$.

   Let $\operatorname{GL}(V)$ denote the group $$\operatorname{GL}(V) = \{ \text{all invertible $F$-linear transformations$\phi:V \to VParseError: KaTeX parse error: Expected 'EOF', got '}' at position 1: }̲\}$ where the group operation is composition of linear transformations.

   Recall that $\operatorname{GL}_n(F)$ denotes the group of all invertible $n \times n$ matrices.

   If $\mathcal{B} = \{b_1,b_2,\cdots,b_n\}$ is a choice of basis, show that the assignment $\phi \mapsto [\phi]_{\mathcal{B}}$ determines an isomorphism $$\operatorname{GL}(V) \xrightarrow{\sim} \operatorname{GL}_n(F).$$

   Here $[\phi]_{\mathcal{B}} = [M_{ij}]$ denotes the matrix of $\phi$ in the basis $\mathcal{B}$ defined by equations

   $$\phi(b_i) = \sum_{k=1}^n M_{ki} b_k.$$