= Fundamental and exterior-square weight systems of type A
{title2=$\varepsilon_i,\quad\varepsilon_i+\varepsilon_j\ (i<j)$}
For the <special unitary group> $SU(N)$, put $\varepsilon_i=e_i-N^{-1}\sum_je_j$. Its defining <representation> has <weight system> $\{\varepsilon_i\}$, and its <exterior square> has <weight system> $\{\varepsilon_i+\varepsilon_j:i<j\}$; each has multiplicity one. Their <dual representations> negate the <weights of a representation>. The <Dynkin label> coordinates are $(\varepsilon_i)_r=\delta_{ir}-\delta_{i,r+1}$. These follow by evaluating diagonal traceless generators on coordinate vectors and their <exterior products>.
Under the standard embedding $SU(N-1)\subset SU(N)$, the last defining <weight of a representation> restricts to zero and the others become the defining <weights of a representation> of $SU(N-1)$. Splitting pairs according to whether they contain the last index proves the <branching of a representation> $V_N\downarrow=V_{N-1}\oplus\mathbf1$ and $\Lambda^2V_N\downarrow=\Lambda^2V_{N-1}\oplus V_{N-1}$, including multiplicities.
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