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\begin{document}

\subsection*{PHYS 570/MA 595: Unit 2 Exercises, Due September 25, 2025}
Sometimes important definitions, derivations, and examples will be deferred to the exercises, but they are not meant to be hard. Exercises will be added to this list but no new problems will be added after Thursday September 18.\\

\textbf{Thursday September 11}
\begin{enumerate}
\item \textbf{Check} that the \hl{Hadamard matrix} $H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}$ interwines the permutation representation of $\mathds{Z}_2$ with the direct sum of the trivial and sign representations. \hspace{10pt} (2 points) 

\item Is it possible for a group representation $V$ to satisfy $V \otimes V = 1 \oplus V$? \textbf{Argue} why or why not. \hfill (2 points) \\


 
\end{enumerate}

\noindent \textbf{Tuesday September 16}
\begin{enumerate}
\item[3.] \textbf{Describe} all multiplicity-free fusion rings of rank 3 which contain a non-self dual basis element. \hfill (6 points)

\item[4.] \textbf{Compute} the Frobenius-Perron dimensions of the \hl{Haagerup} fusion rules on the label set $L = \{1, \omega, \omega^*, X, Y, Z\}$. 

$$\begin{array}{c||c|c|c|c|c|c}
\times & 1 & \omega & \omega^* & X & Y & Z \\
\hline
1 & 1 &  \omega & \omega^* & X & Y & Z \\
\hline
\omega & \omega & \omega^* & 1 & Y & Z & X \\
\hline
\omega^* & \omega^* & 1 & \omega & Z & X & Y \\
\hline
X & X & Z & Y & 1+ X + Y + Z & \omega^* + X + Y + Z & \omega + X + Y + Z \\
\hline
Y & Y & X & Z & \omega + X + Y + Z & 1 + X +Y + Z & \omega^* + X + Y + Z \\
\hline
Z & Z & Y & X &  \omega^* + X + Y + Z & \omega + X+ Y+ Z & 1 + X + Y + Z 
 \end{array}$$

You'll notice this fusion table has some patterns, what can you say about it? \hfill (6 points)
\item[5.] 


\begin{enumerate}

\item[(a)] Say we have two fusion rings, with label sets $L_1, L_2$ and fusion coefficients $N^{a_1b_1}_{c_1}$ for $a_2,b_2,c_2 \in L_2$ and $N^{a_2b_2}_{c_2}$ for $a_2,b_2,c_2 \in L_2$, respectively. \textbf{Check} that $L=L_1 \times L_2$ -- which we will write as $L = \{a \boxtimes b\}_{a \in L_1, b \in L_2}$ -- gives the free $\mathds{Z}$-module on $L$ the structure of a fusion ring with respect to the multiplication on $L$ given by
$(a_1 \boxtimes a_2) \times (b_1 \boxtimes b_2) = \sum_{c_1 \in L_1, c_2 \in L_2} N^{a_1b_1}_{c_1}N^{a_2b_2}_{c_2} c_1 \boxtimes c_2$. \hfill (4 points)

We will see that this categorifies to to a product of categories called the \hl{Deligne product}, and that this operation can be interpreted as a \hl{stacking} of physical theories.

\item[(b)] \textbf{Fill in the blanks} in the fusion table for the fusion ring Fib $\boxtimes$ Ising.  \hfill (2 points)

\begin{tabular}{c||c|c|c|c|c|c}
   & $ 1 \boxtimes 1$ & \hspace{12pt} $1 \boxtimes \sigma$ \hspace{12pt} & \hspace{12pt} $1 \boxtimes \psi$\hspace{12pt} & \hspace{12pt} $ \tau \boxtimes 1$\hspace{12pt}  & \hspace{12pt} $\tau \boxtimes \sigma$\hspace{12pt}  &\hspace{12pt}  $\tau \boxtimes \psi$ \hspace{12pt} \\ \hline \hline  $ 1 \boxtimes 1$ & $ 1 \boxtimes 1$ & $1 \boxtimes \sigma$ & $1 \boxtimes \psi$& $ \tau \boxtimes 1$ & $\tau \boxtimes \sigma$ & $\tau \boxtimes \psi$ \\ \hline
  $1 \boxtimes \sigma$ & $1 \boxtimes \sigma$ &  & & & \\
   \hline
   $1 \boxtimes \psi$ & $1 \boxtimes \psi$ & &  & & & \\
   \hline
   $\tau \boxtimes 1$ & $\tau \boxtimes 1$ & & & & & \\
      \hline
   $\tau \boxtimes \sigma$ & $\tau \boxtimes \sigma$ & & & & & \\
    \hline
   $\tau \boxtimes \psi$ & $\tau \boxtimes \psi$ & & & & & \\
\end{tabular}

\item[(c)] \textbf{Explain} why $d_{a \boxtimes b}= d_ad_b$ in any Deligne product of fusion rings, where $d_a$ is the Frobenius-Perron dimension.\footnote{Recall from lecture that this notation $d_a$ for $a \in L$ is typically reserved for the quantum dimension of an object (rather than the Frobenius-Perron dimension in either a unitary or spherical fusion category but that we're co-opting it anyway.} \hfill (2 points)

\end{enumerate}
\end{enumerate}
\noindent \textbf{Thursday September 18}

\begin{enumerate}

\item[6.] \textbf{Enumerate} the admissible labelings of this diagram using the Ising fusion rules. 
Assume all edges are oriented going down the page. \hfill (3 points)

$$\begin{tikzpicture}[line width=1,baseline=-12.5, scale=.5]
\draw (0,0) node[above] {$\sigma$} -- (2.5,-2.5)--(2.5,-3.5) node[below] {};
\draw (1,0) node[above] {$\sigma$} -- (.5,-.5);
\draw (2,0) node[above] {$\sigma$} -- (1,-1);
\draw (3,0) node[above] {$\sigma$} -- (4,-1);
\draw (4,0) node[above] {$\sigma$} -- (4.5,-.5);
\draw (5,0) node[above] {$\sigma$} -- (2.5,-2.5);
\begin{scope}[yshift=-7cm,yscale=-1]
\draw (0,0) node[below] {$\sigma$} -- (2.5,-2.5)--(2.5,-3.5) node[above] {};
\draw (1,0) node[below] {$\sigma$} -- (.5,-.5);
\draw (2,0) node[below] {$\sigma$} -- (1,-1);
\draw (3,0) node[below] {$\sigma$} -- (4,-1);
\draw (4,0) node[below] {$\sigma$} -- (4.5,-.5);
\draw (5,0) node[below] {$\sigma$} -- (2.5,-2.5);
\end{scope}
\end{tikzpicture}$$



\item[7.] Let $G=S_3$. Enumerate the pairs of the form $(C,\chi)$, where $C$ is a conjugacy class in $G$ and $\chi$ is an irrep of the \hl{centralizer subgroup} $Z(c)=\{g \in G | gc=cg\}$ for a representative $c \in C$. \hfill (3 points)

For example, take $C= \{(123),(132)\}$ and $c=(123)$. Then $Z(c)= \langle (123) \rangle \cong \mathds{Z}_3, \ldots$ and so on.
\end{enumerate}
\end{document}



