Jtdylktuy

Let $m$ and $n$ be integers, with $n>0$ and $gcd(m,n)=1$. Let $theta=m/n$ and let $A_{theta}$ be the rational rotation C$^{*}$-algebra generated by two unitaries $u$ and $v$, satisfying the relation $vu=e^{2pi i theta}uv$. I am working on an exercise in Davidson's C$^{*}$-algebra and my goal is to

Find all irreducible representations of $A_{theta}$ and show that they lie in $M_{n}(mathbb{C})$.

I was able to show that $u^{n}$ and $v^{n}$ lie in the center of $A_{theta}$. Thus, if $picolon A_{theta}to B(H)$ is an irreducible representation of $A_{theta}$, it must be that $pi(u^{n})$ and $pi(v^{n})$ are scalar multiples of the identity. Using this I can show that the set $S={pi(u)^{j}pi(v)^{k}:0leq j,kleq n-1}$ linearly spans $pi(A_{theta})$. Thus, $dim(pi)leq n$. But I don't know how to rule out the case that $dim(pi)< n$ or how to go about finding all of the irreps.

Suppose that $pi(A_theta)subset M_k(mathbb C)$ is a unitary, where $k<n$. Then we have matrices $U,V$ such that $VU=e^{2pi itheta}UV$. In particular,
$$
V=U^*(e^{2pi i theta} V)U.
$$
So $V$ is unitarily equivalent to $e^{2pi i theta}V$. Take $lambdainsigma(V)$. Then
$$
lambda, e^{2pi i theta}lambda, e^{2pi i 2theta}lambda, ldots, e^{2pi i (n-1)theta}lambda
$$
are $n$ eigenvalues of $V$. But $V$ has at most $n-1$ distinct eigenvalues, so there exist integers $r,s$ with $0leq r,sleq n-1$ and $e^{2pi i rtheta}lambda=e^{2pi i stheta}lambda$. This implies $e^{2pi i (r-s)theta}=1$, a contradiction since $|r-s|<n$.