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\begin{document}

\begin{center}
{\bf Math 100C: Spring 2017 \\ Homework 1 \\ Due 4/12/2015}
\end{center}





\vspace{0.1in}
\noindent
{\bf Problem 1:}  
Let $G$ be a finite group and let $R: G \rightarrow GL_1(k)$ be a 1-dimensional representation of $G$.
Prove that the image $R(G)$ is a cyclic group.

\vspace{0.1in}
\noindent
{\bf Problem 2:}  
Let $k$ be a field of characteristic zero and let $V \subset k^n$ be the subspace
$$
V := \{(c_1, \dots, c_n) \in k^n \,:\, c_1 + \cdots + c_n = 0 \} 
$$
where coordinates sum to zero.  The symmetric group $S_n$ acts on $V$ by permutation of coordinates:
$$
\sigma.(c_1, \dots, c_n) = (c_{\sigma(1)}, \dots, c_{\sigma(n)})
$$
for all $\sigma \in S_n$.  Prove that $V$ is an irreducible representation of $S_n$.

\vspace{0.1in}
\noindent
{\bf Problem 3:} 
Let $G$ be a group, let $R: G \rightarrow GL_n(k)$ be an $n$-dimensional matrix representation of $G$,
and let $S: G \rightarrow GL_1(k)$ be a $1$-dimensional matrix representation of $G$.
Prove that the map $R': G \rightarrow GL_n(k)$ given by $R'_g := S_g R_g$ is a representation of $G$.
Prove that $R$ is irreducible if and only if $R'$ is irreducible.



\vspace{0.1in}
\noindent
{\bf Problem 4:}  
Let $G$ be a finite group and let $R: G \rightarrow GL_n(\CC)$ be a complex representation of $G$.
For any $g \in G$, prove that $R_g$ is a diagonalizable matrix.


\vspace{0.1in}
\noindent
{\bf Problem 5:}
Does the conclusion of Problem 4 hold if we drop the assumption that $G$ is finite?
What if we assume that $G$ is finite, but replace $\CC$ by $\mathbb{F}_p$ (the field with $p$ elements; 
here $p$ is a prime)?


\vspace{0.1in}
\noindent
{\bf Problem 6:}
Let $C_n$ be the cyclic group of order $n$.  Classify the irreducible representations of $C_n$ over $\CC$.


\vspace{0.1in}
\noindent
{\bf Problem 7:}
If $A$ and $B$ are matrices, the {\em tensor product} $A \otimes B$ of $A$ and $B$ is the block matrix
$$
A \otimes B = \begin{pmatrix}
a_{11} B & a_{12} B & \cdots \\
a_{21} B & a_{22} B & \cdots \\
\vdots & \vdots & 
\end{pmatrix}.
$$
Hence if $A$ is $n_A \times m_A$ and $B$ is $n_B \times m_B$, then $A \otimes B$ is 
$(n_A n_B) \times (m_A m_B)$.

Let $G$ and $H$ be groups and let $R: G \rightarrow GL_n(k)$ and $S: H \rightarrow GL_m(k)$ be 
matrix representations of $G$ and $H$.  Prove that the map
$$
R \otimes S: G \times H \rightarrow GL_{nm}(k)
$$
given by $(R \otimes S)_{(g,h)} := R_g \otimes S_h$ for all $(g,h) \in G \times H$ is a representation of the 
product group $G \times H$.  (In particular, why is the matrix $(R \otimes S)_{(g,h)}$ invertible?)
Describe the character $\chi_{R \otimes S}$ of $R \otimes S$ in terms of the characters $\chi_R$
of $R$ and $\chi_S$ of $S$.




\end{document}








