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Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights?

Are these the correct Dynkin labels for the weights of $V$: $$\left(1,0,0,0\right),\left(-1,1,0,0\right),\left(0,-1,1,0\right),\left(0,0,-1,1\right),\left(0,0,0,-1\right)$$

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Nov 28 '18 at 11:36
  • $\begingroup$ Related question by OP: physics.stackexchange.com/q/443748/2451 $\endgroup$ – Qmechanic Nov 28 '18 at 11:36
  • $\begingroup$ See chapter X of this nice book. $\endgroup$ – user178876 Nov 28 '18 at 15:44
  • $\begingroup$ ... and the Susyno package has a function "Weights" which does that automatically. $\endgroup$ – user178876 Nov 28 '18 at 15:49
  • $\begingroup$ Ok, so does that mean we have $\left(0,0,0,1\right),\left(-2,1,0,1\right),\left(-1,-1,1,1\right),\left(-1,0,-1,2\right),\left(-1,0,0,0\right)$ for $\Lambda ^2 V$? $\endgroup$ – Joshua Tilley Nov 28 '18 at 16:11
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The simplest way is to recognize that a carrier space for the fully symmetric irrep $(N,0\ldots)$ of SU($m$) is spanned by the harmonic oscillator bosonic states of $N$ total excitation in $m$ dimension. Denoting these by $\vert n_1,n_2,\ldots,n_m\rangle$, the weight of a state is $(n_1-n_2, n_2-n_3\ldots,n_{m-1}-n_m)$.

Applying this to the fundamental of SU($5$), we have the $5$ basis states $\vert 1,0,0,0,0\rangle$, $\vert 0,1,0,0,0\rangle$, $\vert 0,0,1,0,0\rangle$, $\vert 0,0,0,1,0\rangle$ and $\vert 0,0,0,0,1\rangle$ with respective weights $(1,0,0,0),(-1,1,0,0),(0,-1,1,0),(0,0,-1,1)$ and $(0,0,0 -1)$.

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