# Weights of $SU\left(5\right)$ representation

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 $$V$$: $$\left(1,0,0,0\right),\left(-1,1,0,0\right),\left(0,-1,1,0\right),\left(0,0,-1,1\right),\left(0,0,0,-1\right)$$

• Would Mathematics be a better home for this question? – Qmechanic Nov 28 '18 at 11:36
• Related question by OP: physics.stackexchange.com/q/443748/2451 – Qmechanic Nov 28 '18 at 11:36
• See chapter X of this nice book. – user178876 Nov 28 '18 at 15:44
• ... and the Susyno package has a function "Weights" which does that automatically. – user178876 Nov 28 '18 at 15:49
• 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 $\Lambda ^2 V$? – Joshua Tilley Nov 28 '18 at 16:11

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 $$(0,0,0 -1)$$.