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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 14.2  \\ Spring 2023 \hfill Due April 26}}}

\large

\vspace{1cm}


\begin{exercise}
Let $A$ be a finite abelian group.  For $\chi, \psi \in \widehat{A}$, define their {\em inner product} to be
\[
\langle \chi, \psi \rangle = \frac{1}{|A|} \sum_{a \in A} \chi(a) \overline{\psi(a)},
\]
where for a complex number $z = x + i y$ we define $\overline{z} = x - iy$ (the {\em complex conjugate}).  Use exercise 13.3.3 to prove the {\em orthogonality relations} 
\[
\langle \chi, \psi \rangle = \begin{cases}  1 & \text{ if } \chi= \psi, \\ 0 & \text{ otherwise.} \\ \end{cases}
\]
[{\em Suggestion:}  Show that $|\psi(a)| = 1$ and hence $\overline{\psi(a)} = \psi(a)^{-1} = \psi^{-1}(a)$.]
\end{exercise}


\begin{exercise}
Let $A$ be a finite abelian group of order $n$.  Because $A$ is self-dual, this implies $|\widehat{A}| = n$.  Enumerate $A$ and $\widehat{A}$ in some fashion:
\[
A = \{a_1, a_2, \langle, a_n \}, \,\,\,\widehat{A} = \{\chi_1, \chi_2, \ldots, \chi_n \}. 
\]
The {\em character table} of $A$ is the $n \times n$ matrix
\[
M = \frac{1}{\sqrt{n}}(\chi_i (a_j)),
\]
whose rows (up to the factor of $1/\sqrt{n}$) give all the values of a fixed character.  Use the orthogonality relations of the preceding exercise to prove that
\[
MM^{\ast} = I,
\]
where $M^{\ast} = \overline{M}^t$.
\end{exercise}

\begin{exercise}
Continue using the notation of Exercise 2.  In linear algebra one usually proves that if $P$ and $Q$ are square matrices with the same dimensions and $PQ = I$, then $P= Q^{-1}$.  In other words, if $P$ is a left inverse for $Q$, then it is automatically a two-sided inverse.  Exercise 2 then implies that $M^{\ast} M = I$.  Use this to prove that for any $a, b \in A$ one has the {\em dual orthogonality relations}
\[
\frac{1}{|A|}\sum_{\chi \in \widehat{A}} \chi(a) \overline{\chi}(b) = \begin{cases} 1 & \text{ if } a=b,\\ 0 & \text{ otherwise.} \end{cases}
\]
This dual relationship is an essential ingredient in the proof of Dirichlet's theorem on primes in arithmetic progressions.
\end{exercise}

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