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\begin{document}
\noindent {\bf Physics 251} \hfill {\bf Problem Set 2} \hfill {\bf
Spring 2019}
\myrule
\vskip 0.2in
\centerline{\sl DUE: \ TUESDAY, MAY 7, 2019}
\vskip 0.2in\noindent
\underline{{\sl ALERT}}: You should be ready with an initial choice for a term project
topic on Monday April 29.  Feel free to consult with me on possible choices.
A short written proposal (one paragraph would suffice plus references) is due on Tuesday May 7.
\vskip 0.3in

\noindent
1.  The matrix group O($n$) consists of real orthogonal $n\times n$ matrices ($n$ is a positive integer), and SO($n$) consists of the subgroup of O($n$) matrices
with determinant equal to one.   
\vskip 0.1 in

(a) Show that SO($n$) is a normal subgroup of O($n$).
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(b) If $n$ is odd, show that $\mathbb{Z}_2\iso \{\boldsymbol{I}_n\,,\,-\boldsymbol{I}_n\}$  is a normal subgroup of O($n$), where $\boldsymbol{I}_n$ is the $n\times n$ identity matrix.   Prove that
O($n$) can be written as an internal direct product, ${\rm O}(n)\iso{\rm SO}(n)\otimes\mathbb{Z}_2$.
\vskip 0.1in

(c) Explain why the results of part (b) do not apply to the case of even $n$.  Show that if $n$ is even then O($n$) can be written as a semidirect product,
${\rm O}(n)\iso{\rm SO}(n)\rtimes\mathbb{Z}_2$.   Identify explicitly the subgroup of O($n$) appearing in this semidirect product that is isomorphic to~$\mathbb{Z}_2$.


\vskip 0.4in
\noindent
2. A finite group $G$ can be decomposed into conjugacy classes $\mathcal{C}_k$.
\vskip 0.1in

(a) Construct the set $\mathcal{C}^\prime_k\equiv g\mathcal{C}_k g^{-1}$,
which is obtained by replacing each element $x\in\mathcal{C}_k$ by $gxg^{-1}$.
Prove that $\mathcal{C}^\prime_k=\mathcal{C}_k$.
\vskip 0.1in

(b) Suppose that $D^{(i)}(g)$ is the $i$th irreducible (finite-dimensional) matrix representation of the
finite group $G$.  For a fixed class $\mathcal{C}_k$, prove that
\begin{equation} \label{e1}
\sum_{g\in\mathcal{C}_k} D^{(i)}_{j\ell}(g)=\frac{N_k}{n_i}\chi^{(i)}(\mathcal{C}_k)\delta_{j\ell}\,,
\end{equation}
where $n_i$ is the dimension of the $i$th irreducible representation of $G$,
$N_k$ is the number of elements in the $k$th conjugacy class and $\chi^{(i)}(\mathcal{C}_k)$
is the irreducible character corresponding to the $k$th conjugacy class.
\vskip 0.1in

\noindent
{\sl HINT:} Denoting the sum on the left hand side of eq.~(\ref{e1}) by $A_k$ and using the result of part~(a), prove
that $D^{(i)}(g)A_k=A_kD^{(i)}(g)$ for all $g\in G$.  Then use Schur's second lemma.
\vskip 0.1in

(c) Starting from the completeness result that is satisfied by the matrix elements of the
irreducible matrix representations of $G$ and using the result of part~(b), derive the completeness relation for the
irreducible characters,
$$
\frac{N_k}{O(G)}\sum_i \chi^{(i)}(\mathcal{C}_k)[\chi^{(i)}(\mathcal{C}_\ell)]^\ast=\delta_{k\ell}\,,
$$
where $O(G)$ is the order of the group $G$ (i.e.~the number of elements of $G$), and the sum is taken
over all inequivalent finite-dimensional irreducible representations.
\vskip 0.1in

(d) Using the orthogonality and the completeness relations satisfied by the irreducible
characters, prove that the number of inequivalent irreducible representations of $G$ is equal
to the number of conjugacy classes.

\vskip 0.4in
\noindent
3. Consider the transformations of the triangle that make up the
dihedral group $D_3$.  The elements of this group are
$
D_3=\{e,r,r^2,d,rd,r^2d\}\,,
$
with the group multiplication law determined by the relations $r^3=e$,
$d^2=e$ and $dr=r^2d$, where $e$ is the identity element.  In class, the following
two-dimensional representation
matrices 
for $r,d\in D_n$ were given,
\beq \label{dnrep}
r=\begin{pmatrix} \cos(2\pi/n) & \,\, -\sin(2\pi/n) \\
\sin(2\pi/n) & \,\, \phm\cos(2\pi/n)\end{pmatrix}\,,\qquad\quad
d=\begin{pmatrix} 1 & \,\, \phm 0 \\ 0 & \,\, -1\end{pmatrix}\,.
\eeq
Setting $n=3$, one can construct a two-dimensional matrix representation of $D_3$.
\vskip 0.1in

(a) Consider the six-dimensional function space $W$ consisting of polynomials
of degree~$2$ in two real variables $(x,y)$:
\beq \label{f}
f(x,y)=ax^2+bxy+cy^2+dx+ey+h\,,
\eeq
where $a,b,\dots, h$ are complex constants.  We can view $(a,b,\ldots,h)$ as a six-dimensional
vector that lives in a vector space which is isomorphic to $W$.
If we perform a transformation of $(x,y)$ under $D_3$ 
according to the two-dimensional representation obtained from eq~(\ref{dnrep}) with $n=3$, then
the polynomial $f(x,y)$ given by eq.~(\ref{f}) transforms into another polynomial.  That is, the vector
$(a,b,\ldots,h)$ transforms under $D_3$ according to a six-dimensional representation.  Compute the 
$6\times 6$ matrices that represent the elements of $D_3$.  Determine which 
irreducible representations of $D_3$
are contained in this six-dimensional representation and their corresponding
multiplicities.
\vskip 0.1in

(b) Identify the irreducible invariant subspaces of $W$ under $D_3$.  Check that
your result is consistent with the results of part~(a).
\vskip 0.4in

\noindent
4. Consider the dihedral group $D_4$ treated in problem 3 of Problem Set 1.  
The elements of this group are $D_4=\{e,r,r^2,r^3,d,rd,r^2d,r^3 d\}$
with the group multiplication law determined by the relations $r^4=e$,
$d^2=e$ and $dr=r^3d$, where $e$ is the identity element.
\vskip 0.1in

(a) Write out the conjugacy class multiplication table.
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(b) Determine explicitly the matrices of the regular representation.
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(c) Using the 
two dimensional matrix representation given in \eq{dnrep} with $n=4$, verify that the group multiplication table of $D_4$ is
preserved.  Prove that this representation is irreducible.
\vskip 0.1in

(d)  Construct the character table for the irreducible representations of $D_4$.


\vskip 0.4in
\clearpage
\noindent
5. Suppose that $D$ is an irreducible $n$-dimensional representation of a finite group $G$, and $D^{(1)}$ is a (nontrivial) one-dimensional
representation of $G$.  Prove that the direct product $D\otimes D^{(1)}$ is an irreducible representation of $G$.
%Prove that if $G$ is a finite group, then the 
%direct product of an irreducible representation of $G$ and a
%representation of $G$  of dimension 1 is irreducible.
%\vskip 0.1in
%
\noindent
[{\sl HINT}: Show that for any $g\in G$, $|D^{(1)}(g)|=1$.] 

\vskip 0.4in
\noindent
6.~~(a) Display all the standard Young tableaux of the permutation group
$S_4$.  From this result, enumerate the inequivalent irreducible
representations of $S_4$ and specify their dimensions.
\vskip 0.1in

(b) Show that the normal subgroup
$\{e,(12)(34),(13)(24),(14)(23)\}$ of $S_4$ is isomorphic to $D_2$.
Using this result, prove that $D_3\iso S_4/D_2$.  
\vskip 0.1in

(c) Using the two-dimensional irreducible representation of $D_3$ given in class
and the result of part (b),
construct a two-dimensional representation of $S_4$ and determine its
characters.  Is the latter an \textit{irreducible} representation of $S_4$?
\vskip 0.1in

\noindent
{\sl HINT:} Show that given a normal subgroup $N$ of a group $G$ and a representation $D^{G/N}$ of the factor group $G/N$, one
can construct a representation $D^G$ of the group $G$ by defining $D^G(g)\equiv D^{G/N}(gN)$ for all $g\in G$.
\vskip 0.1in

(d) Using the known one-dimensional representations of $S_4$ and the
results of parts~(a) and (c), construct the character table for the group $S_4$.
Determine any unknown entries in the character table by using the
orthonormality and completeness relations for the irreducible
characters.  Using this technique, all entries of the character
table can be uniquely determined up to a sign ambiguity
in some of the entries.
\vskip 0.1in

(e) [{\sl EXTRA CREDIT}] Resolve the sign ambiguity of part (d).  For example, determine
the matrix representative of the transposition $(1\,\,2)$
in the three-dimensional irreducible representation of
$S_4$.  By taking the trace of this matrix, complete the character table of $S_4$. 

\vskip 0.4in
\noindent
7.~~(a) Verify the following properties of the Pauli matrices 
$\boldsymbol{\vec\sigma}\equiv(\sigma_1\,,\,\sigma_2\,,\,\sigma_3)$:
\beqa
&&\hspace{0.5in}\phantom{ii}(i)~\sigma_i\sigma_j=I\delta_{ij}+i\epsilon_{ijk}\sigma_k
\,,\nonumber \\
&&\hspace{0.5in}\phantom{i}(ii)~\sigma_2\,{\boldsymbol{\vec\sigma}}\,\sigma_2
=-{\boldsymbol{\vec\sigma}}^{\,*}\,,\nonumber \\
&&\hspace{0.5in}(iii)~\exp(-i\theta{\boldsymbol{\hat{n}\cdot\vec{\sigma}}}/2)
=I\cos(\theta/2)
-i{\boldsymbol{\hat{n}\cdot\vec{\sigma}}}\sin(\theta/2)\,,\nonumber
\eeqa
where $I$ is the $2\times 2$ identity matrix.
\vskip 0.1in

(b) In the angle-and-axis parameterization of SO(3), a rotation by an
angle $\theta$ about an axis that points along the
unit vector ${\boldsymbol{\hat n}}$
is represented by
an SO(3) matrix given by $R_{ij}({\boldsymbol{\hat n}},\theta)= 
\exp(-i\theta{\boldsymbol{{\hat n}\cdot \vec J}}\,)_{ij}$,
with $({\boldsymbol{{\hat n}\cdot \vec J}}\,)_{ij}\equiv -i
\epsilon_{ijk}n_k$.  By convention, we assume that $0\leq \theta\leq\pi$,
and the axis $\boldsymbol{\hat n}$ can point in any direction.
Evaluate $R_{ij}$ explicitly by summing the Taylor series of the exponential, and show that
$$R_{ij}({\boldsymbol{\hat n}},\theta)= n_i n_j +
(\delta_{ij}-n_i n_j)\cos\theta-\epsilon_{ijk}n_k\sin\theta\,.$$
%Show that $R_{ij}$ is symmetric under the interchange of $i$ and $j$ if and 
%only if $\theta=0$ or $\pi$.  In the latter case, show that   
\vskip 0.1in

(c) Verify the formula:
$e^{-i\theta{\boldsymbol{\hat n\cdot\vec\sigma}}/2}\,\sigma_j\,
e^{i\theta{\boldsymbol{\hat n\cdot\vec\sigma}}/2}=
R_{ij}({\boldsymbol{\hat n}},\theta)\,\sigma_i\,.$
\vskip 0.1in

(d) The set of matrices $\exp(-i\theta{\boldsymbol{\hat{n}\cdot\vec{\sigma}}}/2)$
constitutes the defining representation of SU(2).  Prove that
this representation is pseudoreal.
[{\sl HINT:} Property (ii) of part (a) is useful here.]


\end{document}
