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\title{Math 122: Project on induced representations}
\author{}
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You may discuss the project with myself, Matt Larson, or with one another. You
may {\textit{not}} show another student your written work.

You may use anything you find in Dummit and Foote, Lang's {\textit{Algebra}} or
anything you find posted on the course web page. You may even find some of the
questions proved verbatim.

By a {\textit{$G$-module}} $V$ we mean a $\mathbb{C} [G]$-module. Thus
$G$-modules are equivalent to representations of $G$. In contrast with
notation used in Lecture~8, here $V^G$ does \textit{not} mean the module
of invariants. It is the induced module.

\bigbreak\noindent
1. Let $G$ be a finite group and let $V, W$ be $\mathbb{C} [G]$ modules, not
necessarily irreducible. Let $\chi_V$ and $\chi_W$ be the characters. Explain
why
\[ \dim \, \Hom_{\mathbb{C} [G]} (V, W) = \langle \chi_V, \chi_W \rangle . \]

\bigbreak\noindent
2. Let $H$ be a subgroup of the finite group $G$. Let $V$ be an $H$-module.
Prove that there exists a $G$-module $V^G$, unique up to isomorphism, such
that for every $G$-module
\[ \label{frorec} \dim \; \Hom_{\mathbb{C} [G]} (V^G, W) = \dim \;
   \Hom_{\mathbb{C} [H]} (V, W) . \]

\medbreak\noindent
{\textbf{Hint:}} Explain why it is sufficient to check this for $W$
irreducible. Let $V_1, \cdots, V_h$ be the distinct irreducible $G$-modules.
Postulate a solution of the form
\[ V^G = \bigoplus d_i V_i . \]
What should the multiplicities $d_i$ be?

\bigbreak\noindent
Problem 2 suggests that the {\textit{induced representation}} $V^G$ should be a
functor from the category of $H$-modules to the category of $G$-modules.

\bigbreak\noindent
3. Let $V$ be an $H$-module. Let $\pi : H \longrightarrow \GL(V)$ be the
corresponding representation, so $\pi (h) v = h v$. Let $I (V) = I_H^G (V)$ be
the space of all maps $f : G \longrightarrow V$ that satisfy
\[ f (h g) = \pi (h)\bigl(f (g)\bigr),\qquad h\in H . \]
Define a map $\Pi : G \longrightarrow \GL (I (V))$ by
\[ \bigl(\Pi (g) f\bigr) (x) = f (x g), \qquad g, x \in G. \]

\medbreak\noindent
(a) Prove that $\Pi (g) f \in I (V)$ if $f \in I (V)$ and $g \in G$.

\medbreak\noindent
(b) Prove that $\Pi$ is a representation of $G$.

\medbreak\noindent
(c) If $v \in V$ define a map $\hat{v} : G \longrightarrow V$ by
\[ \hat{v} (g) = \left\{\begin{array}{ll}
     g v & \text{if $g \in H$,}\\
     0 & \text{otherwise.}
   \end{array}\right. \]
Prove that $\hat{v} \in I (V)$ and that the map $\alpha : V \longrightarrow I
(V)$ defined by $\alpha (v) = \hat{v}$ is an $H$-module homomorphism.

\medbreak\noindent
(d) Prove the following {\textit{universal property}}: if $\theta : V
\longrightarrow W$ is any $H$-module homomorphism into a $G$-module $W$ then
there is a unique $G$-module homomorphism $\Theta : I (V) \longrightarrow W$
such that $\theta = \Theta \alpha$.

\medbreak\noindent
(e) Prove that $I (V) \cong V^G$ as a $G$-module.

\bigbreak\noindent
4. Let $\chi$ be the character of the $H$-module $V$ and let $\dot{\chi} : G
\longrightarrow \mathbb{C}$ be the function
\[ \dot{\chi} (g) = \left\{\begin{array}{ll}
     \chi (g) & \text{if $g \in H$,}\\
     0 & \text{otherwise.}
   \end{array}\right. \]
Prove that the character $\chi^G$ of the induced representation $V^G = I (V)$
is given by the formula
\[ \chi^G (x) = \frac{1}{| H |} \sum_{g \in G} \dot{\chi} (g x g^{- 1}) . \]


\bigbreak\noindent
5. If $f$ is a class function on $G$ that can be written as a difference of
two characters then $f$ is called a {\textit{virtual character}} or a
{\textit{generalized character}}.

\medbreak\noindent
(a) Explain briefly why the generalized characters form a ring, called the
{\textit{character ring}}.

\medbreak\noindent
(b) Suppose that $f$ is a generalized character and
\[ \langle f, f \rangle = 1, \qquad f (1) > 0. \]
Prove that $f$ is the character of an irreducible representation.

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