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\usepackage{amsfonts}         
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%\usepackage[numbers]{natbib}

\usepackage{enumitem}

\usepackage{slashed}




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\def\gG{\mathsf{G}}
\def\ii{{\bf i}}
\def\Ione{\mathbb{I}}
\def\UU{{\bf U}}
\def\HH{{\bf H}}
\def\pp{{\bf p}}
\def\aa{{\bf a}}
\def\qq{{\bf q}}
\def\eps{\epsilon}
\def\half{{1\over 2}}
\def\Tr{{{\rm Tr~ }}}
\def\tr{{\rm tr}}
\def\Re{{\rm Re\hskip0.1em}}
\def\Im{{\rm Im\hskip0.1em}}
\def\ppi{\boldsymbol{\pi}}
\def\pphi{\boldsymbol{\phi}}
%\def\pphi{\phi}
\def\grad{\vec \nabla}
\def\vB{\vec B}
\def\vE{\vec E}
\def\vA{\vec A}
\def\vAA{ \vec{\bf A}}
\def\vEE{{{\vec {\bf E}}}}
\def\vBB{{\vec {\bf B}}}

\def\CA{{\cal A}}
\def\CB{{\cal B}}
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\def\CD{{\cal D}}
\def\CE{{\cal E}}
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\def\CJ{{\cal J}}
\def\CK{{\cal K}}
\def\CL{{\cal L}}
\def\CM{{\cal M}}
\def\CN{{\cal N}}
\def\CO{{\cal O}}%AEL
\def\CP{{\cal P}}
\def\CQ{{\cal Q}}
\def\CR{{\cal R}}
\def\CS{{\cal S}}
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\def\CV{{\cal V}}
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%%                              TABLEAUX.TEX
%%      This  macro file is for producing a ``Young Tableau'' which is
%%      an array of little squares sometimes used in mathematical physics.
%%      For instance, the command $\tableau{6 3 2}$ will produce a tableau
%%      with 6 squares in the top row, 3 in the next, and 2 in the last.
%%                                  OOOOOO
%%      This tableau will look like OOO    but made of squares instead of O's.
%%                                  OO
%%      Any number of rows may be present, each having a nonzero number of
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%%
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%%                                                      -- Doug Eardley 9/19/8%%
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\def\gettableau#1 {\ifnum#1=0\let\next=\null\else
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%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
%pdftex, 
bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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linkcolor=BrickRed, 
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%                                %
%                                % to use this, wrap text with the conditional, eg:
%                                %
%                                % \ifeq
%                                % SHOW THIS IFF \eqtrue HAS BEEN DECLARED
%                                % \fi
%
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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 6 \answer{--~~~ Solutions} }
\end{center}




\noindent
\hfill {\bf Due 11:59pm { Thursday, November 7, 2024}} 


\hrule


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


\begin{enumerate}

\item {\bf Fun with the Frobenius-Schur indicator.}

\begin{enumerate}


\item Compute the Frobenius-Schur indicator for the ${\bf 2}$ of $S_3$ using the matrices you constructed on the last homework.
If you get zero, can you find the basis where it is real?


\item Consider the 2-dimensional representation of $Q_8$. Using the character table,
decide if it is complex, real or pseudoreal.
%is pseudoreal using the character table.





Write down its representation matrices in terms of Pauli matrices and check your answer.



\item Let $ \sigma_h \equiv $ the number of square roots $g$ of the element $h = g^2$ in the group $G$.
Then by writing the Frobenius-Schur indicator for an irrep as 
\be\label{eq:FS} \eta_a \equiv{1\over |G| } \sum_{g \in G} \chi_a(g^2)  = {1\over |G| } \sum_{h = g^2 \in G} \sigma_h \chi_a(h) \ee
and using character orthogonality, 
show that 
\be\label{eq:answer} \sigma_h = \sum_{\text{irreps}, a} \eta_a \chi_a(h) . \ee



\end{enumerate}


\item 
{\bf Fun with Frobenius reciprocity.}

\begin{enumerate}
\item{} Take each irrep of $S_3$ and decompose it into irreps of $\IZ_3 \subset S_3$ (for example using the characters).  



\item For each of the irreps of $\IZ_3$, construct the induced representation of $S_3$
(you did this already for the trivial representation)
and decompose it into irreps of $S_3$ (for example using the characters).


\item Check Frobenius reciprocity.  



\item{} [Bonus problem] Verify Frobenius reciprocity for $A_4 \subset S_4$.  


\end{enumerate}



%\end{enumerate}

%\item {\bf Normal modes.} 
%
%
%\cor
%\item {\bf Brain-warmer.}
%$$ P_a = {d_a \over |\gG|} \sum_{g \in \gG} \chi_a(g)^\star D(g) .$$
%This operator projects onto $\underbrace{R_a \oplus R_a \oplus \cdots }_{m_a \text{ times}}$.
%Check that these projectors are orthogonal $P_a P_b = \delta_{ab} P_a$ by virtue of the character orthogonality.
%\cob


\item {\bf $A_5$ is simple.}

\begin{enumerate}
\item Show that an invariant subgroup $H \subset G$ is a union of conjugacy classes of $G$.



\item{} [Bonus problem] Find the number of elements of the conjugacy classes $n_C$ of $A_5$.
The answer is 
$n_C = 1,15,20,12,12$. 



\item Show that $A_5$ is simple by checking for possible sums of $n_C$s (necessarily including 1) that divide $5!/2$.

%The proper divisors of $60$ are $2,3,5,6,10,12,15,20,

\end{enumerate}

\item {\bf Induced representations.}
\begin{enumerate}
\item Check that the definition of induced representation in the notes 
actually produces a representation of $G$ in the sense that 
$D(g_1 g_2) \ket{n,x} = D(g_1) D(g_2) \ket{n,x}$ 
for all $g_{1,2} \in G$ and $\ket{n,x} \in W \times V_{G/H}$.


\item Show that the character of the induced representation $\text{Ind}_H^G(W)$ is 
$$ \chi_{\text{Ind}_H^G(W)}(g) = \text{Ind}_H^G[\chi_W](g) $$
where the operation on the RHS is defined in the lecture notes.


\item Prove Frobenius reciprocity: 
$$ \vev{\psi, \text{Res}_H^G[\phi]}_H = \vev{ \text{Ind}_H^G[\psi], \phi}_G. $$

\end{enumerate}



\end{enumerate}

\end{document}
