Let $n$ be a positive integer and $U(n)$ be the group of $n\times n$ unitary matrices. I have two question regarding the irreducible unitary representations of $U(n)$.
Is there any irreducible unitary representation $\pi$ of $U(n)$ such that $1<\dim \pi < n$, where $\dim \pi$ is the dimension of the corresponding representation space.
Let $\pi_1:U(n)\to GL_n(C)$ and $\pi_2:U(n)\to GL_n(C)$ be defined by $$\pi_1(u)=u,~~~\pi_2(u)=\overline u,~u\in U(n).$$ Here $\overline{u}$ is the conjugate matrix of $u$. Clearly, $\pi_1$ and $\pi_2$ are two irreducible unitary representation of $U(n)$ where the dimension of the corresponding representation space is $n$. Are these all irreducible unitary representation of $U(n)$ (upto unitary equivalence) where the dimension of the corresponding representation space is $n$?
I know that there is a Weyl dimension formula for irreducible unitary representation of $U(n)$. But I was not able to solve the above questions using that formula. Also I am very new in this area. Any help or reference will be highly appreciated.
Thanks in advance!