# Characters of the fundamental representations of $SU(3)$

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).