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Sum of weights of an irreducible representation of $U(N)$

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. … Note that we are considering $U(N)$ rather $SU(N)$, because I think the sum of weights of an irrep vanishes for $SU(N)$. …
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