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%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
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bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 220 Symmetries Fall 2024}\\
{\Large\bf Assignment 3 \answer{--~~~ Solutions} }
\end{center}

Comment: this problem set contains many problems, but most of them are simple and many of them are bonus problems.    


\noindent
\hfill {\bf Due 3:30pm { Thursday, October 17, 2024}} 


\hrule




\begin{enumerate}

\item{\bf Brain-warmer.} 
\begin{enumerate}
\item Consider the action of a group $ \gG$ on itself by left multiplication.  
What kinds of terms are allowed in the cycle indicator for this group action?

Compute the cycle indicators for the action by left multiplication of $Q_8$, of $D_4$ and of $\IZ_n$ (do the case of $n$ prime first, and do general $n$ as a bonus problem).
What do you get if you set all the $z_i = 1$?

\def\jj{{\bf j}}
\def\kk{{\bf k}}




\item 
Find the cycle index for the action of $Q_8$ on itself by conjugation.


\end{enumerate}

\item{\bf Brain-warmer.} 
Given a group homomorphism $ \phi: G \to K$, 
show that its kernel $H \equiv \ker \phi \equiv \{ g \in G | \phi(g) = e \in K\} \subset G$
is always a subgroup.
Further, show that it is a {\it normal} subgroup.


\item{\bf Representation on cosets.} Given a group $G$ and a subgroup $H$, we can construct a representation of $G$
by its action on the cosets $x \in G/H$.  
The action is
$$ \begin{matrix}\mathmap{G/H}{G/H}{x=\{g_1, g_2 \cdots \}}{\{gg_1, gg_2 \cdots \} }\end{matrix}.$$
To see more explicitly what this does, choose a representative $x_i$ of each distinct coset.
Then $G = \cup_{i} x_i H$, and for each $g \in G$, $g x_i H $ is again a coset and therefore equal to one of the $x_i H$.
So left multiplication by $g$ permutes the cosets.
\footnote{A word of clarification here: an action of a group $G$ on a set
(recall that for a set $X = \{x_1, .. x_k\}$ of order $k$, this is a homomorphism from $G \to S_k$)
is also a representation in the following sense:
define the carrier space $V = \text{span}\{ \ket{x_i}, i=1..k \} $, and define the linear operators to act on the basis vectors by
$ D(g) \ket{x} = \ket{gx} $.
This is a special kind of representation called a permutation representation, since
the linear operators take basis vectors to basis vectors, rather than making linear combinations -- the matrix elements in this basis are a single 1 in each row and column and zeros in all the other entries.
}

Consider the case where $H = \langle (123) \rangle$ is the subgroup of $S_3$ generated by the order-3 element $(123)$.  
First decompose $S_3$ into cosets by $H$.
Write out the matrices in a basis.
What representation of $S_3$ do you get by the construction above?  (If it is reducible, decompose it into irreps, for example by computing its character.)



What about the case where $H' = \langle(12)\rangle$?  In each case, decompose it into irreps.


This construction plays an important role in the idea of induced representations.
We will learn to call it the representation of $G$ induced by the trivial representation of $H$.


\item {\bf The class of inverses.} 
Given a conjugacy class $c$, define the class $\bar c$ to consist of the inverses of each of the elements in $c$.  
Convince yourself that this is well-defined.
Show that for a unitary representation $R$,
$$ \chi_R(\bar c) = \chi_R(c)^\star .$$



\item {\bf Character exercise.}
Recall the definition of the regular representation of a finite group $\gG$: 
$$ \CH_\gG \equiv \text{span} \{ \ket{g}, ~g \in \gG\}.$$
This space also carries a representation of $\gG \times \gG$ by
$$ \Gamma(m,n) \ket{h} = \ket{m h n^{-1}} ,~ (m,n) \in \gG \times \gG. $$
$\CH_\gG$ is also reducible as a representation of $\gG\times \gG$.  Show that
$$ \CH_\gG = \oplus_{R, \text{irreps of }\gG} R \otimes \bar R $$
where $ \bar R$ is the conjugate representation to $R$, with $\Gamma_{\bar R}(g) = \Gamma_R(g)^\star$.

Hint: Find the characters of $\CH_\gG$ as a representation of $\gG \times \gG$, and 
compute $ \vev{ \chi_{\CH_\gG} | \chi_{R_i \otimes R_j}  } $, 
where $R_i$ are all the irreps of $\gG$.



\item {\bf Reps from characters.}  Using the character table, make unitary representation matrices for the 2-dimensional representation of $D_3 = S_3$
in which $D(123)$ and $D(132)$ are diagonal.  Note that $ 1 + \omega + \omega^2 = 0 $ where $\omega \equiv e^{ 2\pi\ii/3}$.




\end{enumerate}

\end{document}
