Questions about root operators.

Question

I am reading the notes. On line 13 in the section Root operators, it is said that The operator $f_1$ maps from the space $V(\mu)$ to $V(\mu-(1,-1,0))$. I don't know why. We have $$ f_i V (\mu) \subseteq V (\mu - \alpha_i). $$ In $GL_3$, we have $\alpha_1=2\omega_1-\omega_2$. I think that $f_1$ maps from the space $V(\mu)$ to $V(\mu-(2,-1,0))$. Why $f_1$ maps from the space $V(\mu)$ to $V(\mu-(1,-1,0))$? Thank you very much.

Answer 1

We have $\omega_1=(1,0,0)$, $\omega_2=(1,1,0)$. Therefore $2\omega_1-\omega_2=(1,-1,0)$.