Let us denote $3$ and $\bar{3}$ the fundamental representations of $SU(3)$. According to my lecture notes, the characters read as follows:

$\chi_{[3]} = e^{\omega_1} + e^{\omega_1 - \alpha_1} + e^{\omega_1 - \alpha_1 - \alpha_2}$

$\chi_{[\bar{3}]} = e^{\omega_2} + e^{\omega_2 - \alpha_1} + e^{\omega_2 - \alpha_1 - \alpha_2}$

How does one derive that result?

I know that it is somehow related to the Weyl-character formula, and that $\omega_1, \omega_2$ are the weights and $\alpha_1 , \alpha_2$ are the roots, but I am having big trouble understading those concepts and applying them to a specific example.

(I would also appreciate a clarification on which are the fundamental representations of $SU(3)$, i.e., what do exactly $3$ and $\bar{3}$ stand for).


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