I have just started to learn about the representation theory of $\mathfrak{gl}_3(\Bbb C)$ and am confused.
I have read of the fundamental weights $$\omega_1=(1,0,0),\omega_2=(1,1,0),\omega_3=(1,1,1)$$ and know that these correspond to the representations: $$\Bbb C^3,\wedge^2\Bbb C^3,-$$
Q1. Does $\mathfrak{gl}_{n+1}(\Bbb C)$ have only $n$ fundamental reps, although it has $n+1$ fundamental weights [where $\mathfrak{sl}_{n+1}(\Bbb C)$ has one fewer since $\epsilon_1+\dots+\epsilon_n=0$] - Then $\mathfrak{sl}_{n+1}$ and $\mathfrak{sl}_n$ have the same representation spaces for the fundamental reps?
Q2. Do all finite dimensional highest weight representations of $\mathfrak{gl}_3(\Bbb C)$ arise as subrepresentations of a tensor product of the fundamental representations?
Q3. Then, in terms of the fundamental representations, where does the irreducible representation corresponding to the dominant weight $\lambda=(2,1,0)$ live. i.e $V(2,1,0)$.
A guess would be that since $\lambda = \omega_1+\omega_2$, perhaps it lives in $\Bbb C^3\otimes \wedge^2\Bbb C^2$? I believe that two highest weight representations $V,W$ with highest weights $\lambda,\mu$ respectively make $V\otimes W$ have highest weight $\lambda+\mu$?