ProblemSet 3 -- representation theory

Work any 3 of the following 4 problems.

In these exercises, $G$ always denotes a finite group. Unless indicated otherwise, all vector spaces are assumed to be finite dimensional over the field $F=\mathbb{C}$. The representation space $V$ of a representation of $G$ is always assumed to be finite dimensional over $\mathbb{C}$.

1. Let $\varphi :G\to F^{\times }$ be a group homomorphism; since $F^{\times }={\mathrm{GL}}_1(F)$, we can think of $\varphi$ as a 1-dimensional representation $(\varphi ,F)$ of $G$.

   If $V$ is any representation of $G$, we can form a new representation $\varphi \otimes V$. The underlying vector space for this representation is just $V$, and the “new” action of an element $g\in G$ on a vector $v$ is given by the rule $$g\star v=\varphi (g)gv.$$

   1. Prove that if $V$ is irreducible, then $\varphi \otimes V$ is also irreducible.

   2. Prove that if $\chi$ denotes the character of $V$, then the character of $\varphi \otimes V$ is given by $\varphi \cdot \chi$; in other words, the trace of the action of $g\in G$ on $\varphi \otimes V$ is given by $${\chi }_{\varphi \otimes V}(g)=\mathrm{tr}(v\mapsto g\star v)=\varphi (g)\chi (g).$$

   3. Recall that in class we saw that $S_3$ has an irreducible representation $V_2$ of dimension 2 whose character ${\psi }_2$ is given by

      $$\begin{array}{llll}g& 1& (12)& (123)\\ {\psi }_2& 2& 0& -1\end{array}$$

      Observe that $\mathrm{sgn}\psi =\psi$ and conclude that $V_2\simeq \mathrm{sgn}\otimes V_2$, where $\mathrm{sgn}:S_n\to \{\pm 1\}\subset {\mathbb{C}}^{\times }$ is the sign homomorphism.

      On the other hand, $S_4$ has an irreducible representation $V_3$ of dimension 3 whose character ${\psi }_3$ is given by

      $$\begin{array}{llllll}g& 1& (12)& (123)& (1234)& (12)(34)\\ {\psi }_3& 3& 1& 0& -1& -1\end{array}$$

      (I’m not asking you to confirm that ${\psi }_3$ is irreducible, though it would be straightforward to check that $⟨{\psi }_3,{\psi }_3⟩=1$).

      Prove that $V_3\not\simeq \mathrm{sgn}\otimes V_3$ as $S_4$-representations.

      (In particular, $S_4$ has at least two irreducible representations of dimension 3.)

2. Let $V$ be a representation of $G$.

   For an irreducible representation $L$, consider the set $$\mathcal{S}=\{S\subseteq V\mid S\simeq L\}$$ of all invariant subspaces that are isomorphic to $L$ as $G$-representations.

   Put $$V_{(L)}=\sum _{S\in \mathcal{S}}S.$$

   1. Prove that $V_{(L)}$ is an invariant subspace, and show that $V_{(L)}$ is isomorphic to a direct sum $$V_{(L)}\simeq L\oplus \cdots \oplus L$$ as $G$-representations.

   2. Prove that the quotient representation $V/V_{(L)}$ has no invariant subspaces isomorphic to $L$ as $G$-representations.

   3. If $L_1,L_2,\cdots ,L_m$ is a complete set of non-isomorphic irreducible representations for $G$, prove that $V$ is the internal direct sum $$V=\underset{i=1}{\overset{m}{⨁}}{V}_{(L_i)}.$$

3. Let $\chi$ be the character of a representation $V$ of $G$. For $g\in G$ prove that $\overline{\chi (g)}=\chi (g^{-1})$.

   Is it true for any arbitrary class function $f:G\to \mathbb{C}$ that $\overline{f(g)}=f(g^{-1})$ for every $g$? (Give a proof or a counterexample…)

4. For a prime number $p$, let $k={\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}$ be the field with $p$ elements. Let $V$ be an $n$-dimensional vector space over ${\mathbb{F}}_p$ for some natural number $n$, and let $$⟨\cdot ,\cdot ⟩:V\times V\to k$$ be a non-degenerate bilinear form on $V$.

   (A common example would be to take $V={\mathbb{F}}_{p^n}$ the field of order $p^n$, and $⟨\alpha ,\beta ⟩={\mathrm{tr}}_{{\mathbb{F}}_{p^n}/{\mathbb{F}}_p}(\alpha \beta )$ the trace pairing).

   Let us fix a non-trivial group homomorphism $\psi :k\to {\mathbb{C}}^{\times }$ (recall that $k=\mathbb{Z}/p\mathbb{Z}$ is an additive group, while ${\mathbb{C}}^{\times }$ is multiplicative). Thus $$\psi (\alpha +\beta )=\psi (\alpha )\psi (\beta )\text{for all}\alpha ,\beta \in k.$$ If you want an explicit choice, set $\psi (j+p\mathbb{Z})=\exp (j\cdot 2\pi i/p)=\exp (2\pi i/p{)}^j.$

   For a vector $v\in V$, consider the mapping ${\mathrm{\Psi }}_v:V\to {\mathbb{C}}^{\times }$ given by the rule $${\mathrm{\Psi }}_v(w)=\psi (⟨w,v⟩).$$

   1. Show that ${\mathrm{\Psi }}_v$ is a group homomorphism $V\to {\mathbb{C}}^{\times }$.

   2. Show that the assignment $v\mapsto {\mathrm{\Psi }}_v$ is injective (one-to-one).

      (This assignment is a function $V\to \mathrm{Hom}(V,{\mathbb{C}}^{\times })$. In fact, it is a group homomorphism. Do you see why? How do you make $\mathrm{Hom}(V,{\mathbb{C}}^{\times })$ into a group?)

   3. Show that any group homomorphism $\mathrm{\Psi }:V\to {\mathbb{C}}^{\times }$ has the form $\mathrm{\Psi }={\mathrm{\Psi }}_v$ for some $v\in V$.

      Conclude that there are exactly $|V|=q^n$ group homomorphisms $V\to {\mathbb{C}}^{\times }$.