Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.

Firstly, I would like to know if it is true that $$\sum_{\mu\in W_R}\mu=A(R)(1,\ldots,1),$$ where $A(R)$ is some integer that varies with the representation. Note that we are considering $U(N)$ rather $SU(N)$, because I think the sum of weights of an irrep vanishes for $SU(N)$.

Secondly, I would like to know if there is a simple formula for $A(R)$, or if there is an existing name for it?

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.

Firstly, I would like to know if it is true that $$\sum_{\mu\in W_R}m_\mu\mu=A(R)(1,\ldots,1),$$ where $m_\mu$ is the multiplicity of the weight $\mu$ and $A(R)$ is some integer that varies with the representation. Note that we are considering $U(N)$ rather $SU(N)$, because I think the sum of weights of an irrep vanishes for $SU(N)$.

Secondly, I would like to know if there is a simple formula for $A(R)$, or if there is an existing name for it?