PS02--rep-theory.md

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title: |
 ProblemSet 2 -- Representations and characters
author: George McNinch
date: 2024-01-29

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\newcommand{\trivial}{\mathbf{1}}

In these exercises, $G$ always denotes a finite group and all vector spaces are assumed to be finite dimensional over the field $F = \mathbb{C}$.

In these exercises, you may use results stated but not yet proved in class about characters of representations of $G$.

1. In this problem, we identify the character $\chi_\Omega$ of the permutation representation $(\rho,F[\Omega])$ of a group $G$.

   a. Let $V$ be a vector space and let $\Phi:V \to V$ a linear mapping If $\mathcal{B}$ is a basis for $V$, recall that the trace of $\Phi$ is defined by $$\operatorname{tr}(\Phi) = \operatorname{tr}([\Phi]_{\mathcal{B}}).$$

   apologies -- this is just explanatory; it isn't actually a question

   b. Recall that the dual of $V$ is the vector space $V^\vee = \operatorname{Hom}_F(V,F)$ of linear functionals on $V$.

   If $b_1,\dots,b_n$ is a basis for $V$, let ${b_j}^\vee:V \to F$ be defined by ${b_j}^\vee(b_i) = \delta_{i,j}$. Show that ${b_1}^\vee,\dots,{b_n}^\vee$ is a basis for $V^\vee$; it is known as the dual basis to $b_1,\dots,b_n$.

   c. Prove that the trace of the linear mapping $\Phi:V \to V$ is given by the expression $$\operatorname{tr}(\Phi) = \sum_{i=1}^n {b_i}^\vee(\Phi(b_i)).$$

   d. Suppose that the finite group $G$ acts on the finite set $\Omega$, and consider the corresponding permutation representation $(\rho,F[\Omega])$ of $G$. Recall that $F[\Omega]$ is the vector space of all $F$-values functions on $\Omega$, and that for $f \in F[\Omega]$ and $g \in G$, we have $$\rho(g)f(\omega) = f(g^{-1}\omega).$$ In particular, we saw in the lecture that $$\rho(g)\delta_\omega) = \delta_{g\omega},$$ where $\delta_\omega$ denotes the Dirac function at $\omega \in \Omega$.

   Show that $$\operatorname{tr}(\rho(g)) = \#\{\omega \in \Omega \mid g\omega = \omega\};$$ i.e. the trace of $\rho(g)$ is the number of fixed points of the action of $g$ on $\Omega$.

2. Let $V$ be a representation of $G$, suppose that $W_1,W_2$ are invariant subspaces, and that $V$ is the internal direct sum $$V = W_1 \oplus W_2.$$

   Show that the character $\chi_V$ of $V$ satisfies $$\chi_V = \chi_{W_1} + \chi_{W_2}$$ i.e. for $g \in G$ that $$\chi_V(g) = \chi_{W_1}(g) + \chi_{W_2}(g).$$

3. Let $G = A_4$ be the alternating group of order $\dfrac{4!}{2} = 12$.

   We are going to find the character table of this group.

   a. Confirm that the following list gives a representative for each of the conjugacy classes of $G$:

   $$1, (12)(34), (123), (124)$$

   (Note that $(123)$ and $(124)$ are conjugate in $S_4$, but not in $A_4$).

   What are the sizes of the corresponding conjugacy classes?

   b. Let $K = \langle (12)(34), (14)(23)\rangle$. Show that $K$ is a normal subgroup of index $3$, so that $G/K \simeq \mathbb{Z}/3\mathbb{Z}$.

   Let $\zeta_3$ be a primitive $3$rd root of unity in $F^\times$ and for $i=0,1,2$ let $\rho_i:G \to F^\times$ be the unique homomorphism with the following properties:

   i. $\rho_i\left( (123) \right) = \zeta^i$ ii. $K \subseteq \ker \rho_i$.

   Explain why ParseError: KaTeX parse error: Undefined control sequence: \trivial at position 10: \rho_0 = \̲t̲r̲i̲v̲i̲a̲l̲,\rho_1,\rho_2 determine distinct irreducible (1-dimensional) representations of $G$.

   c. Let $\Omega = \{1,2,3,4\}$ on which $G$ acts by the embedding $A_4 \subset S_4$.

   Compute the character $\chi_\Omega$ of the representation $F[\Omega]$. (This means: compute and list the values of $\chi_\Omega$ at the conjugacy class representatives given in a.)

   (Use the result of problem 1 above).

   d. The span of the vector $\delta_1 + \delta_2 + \delta_3 + \delta_4 \in F[\Omega]$ is an invariant subspace isomorphic to the irreducible representation $\rho_0$ (the so-called trivial representation).

   Thus $F[\Omega] = \rho_0 \oplus W$ for a $3$-dimensional invariant subspace. Explain why problem 2 shows that the character of $W$ is given by ParseError: KaTeX parse error: Undefined control sequence: \trivial at position 24: … \chi_\Omega - \̲t̲r̲i̲v̲i̲a̲l̲.

   Now prove that $\langle \chi_W, \chi_W \rangle = 1$ and conclude that $W$ is an irreducible representation.

   e. Explain why ParseError: KaTeX parse error: Undefined control sequence: \trivial at position 1: \̲t̲r̲i̲v̲i̲a̲l̲,\rho_1,\rho_2,… is a complete set of non-isomorphic irreducible representations of $G$.

   f. Display the character table of $G = A_4$.