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\title{Representation theory \\ Assignment 1 \\ Due Friday January 28}



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\begin{enumerate}
\item Let $ G $ be a finite group acting on a finite set $ X$.  Explain how to construct a representation of $ G $ on $ V = \mathbb{C}[X] $.  Prove that $ \chi_V(g) $ is the number of fixed points of $ g $ acting on $ X $.

\item
Let $ V $ be the 2-dimensional irreducible representation of $ S_3 $.  Using the character table computed in class, decompose $ V^{\otimes n } $ as a representation of $ S_3$.

\item
Find the character table of $ S_4 $.

\item
Let $ G $ be a finite group and $ V $ be an irreducible representation. Prove that the dimension of $ V $ divides the size of $ G $.


\item
Prove that if $ G $ is a finite group, then it is impossible to find a proper subgroup $ T $, such that every element of $ G $ is conjugate into $ T $.  

Use this to prove that if $ G $ is a finite group and $ T $ is a proper subgroup, then the map $ Rep(G) \rightarrow Rep(T) $ (given by restriction of representations) is not injective.

\item
Take $ T = U(1)^2 $, thought of as $2\times 2 $ unitary diagonal matrices.  $T$ acts on $ \mathbb{C}^2 $ in the obvious manner.  Decompose $ (\mathbb{C}^2)^{\otimes n} $ as a representation of $ T $.  (This means find all the weight spaces and their dimensions.)  Do the same thing for $ Sym^n \mathbb{C}^2 $.

\item
Consider $ \mathbb{C}^\times $ and its coordinate ring $$ R = \mathcal{O}(\mathbb{C}^\times) = \mathbb{C}[z, z^{-1}].$$  Define a $\mathbb{C}$-antilinear ring homomorphism $ \sigma : R \rightarrow R $ by setting $ \sigma(z^n) = z^{-n} $, and extending ``antilinearly'', so that
$$
\sigma( \sum_n a_n z^n) = \sum_n \overline{a_n} z^{-n} 
$$
where $ \overline{\phantom{a}} $ denotes complex conjugation.

Prove that $ R^\sigma = \{ f \in R : f^\sigma = f \} $ is isomorphic to $$ \mathbb{R}[x,y]/(x^2 + y^2 -1).$$

Now generalize this result.  If $ T $ is a compact torus and $ T_{\mathbb{C}} $ is its complexification, construct an analog of $ \sigma $ and compute its invariants.

\end{enumerate}

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