Realization of fundamental weights. Let $\mathfrak{gl}_3$ be the general linear Lie algebra. It is said that the fundamental weights $\omega_1$, $\omega_2$ can be realized as vectors $(1,0,0)^T$ and $(1,1,0)^T$ respetively. The simple roots are $\alpha_1=e_1-e_2$, $\alpha_2=e_2-e_3$. The relation between simple roots and fundamental weights is 
\begin{align}
& \alpha_1 = 2\omega_1-\omega_2, \\
& \alpha_2=-\omega_1+2\omega_2,
\end{align}
since $\alpha_i = \sum_{j} C_{ij}\omega_j$, where $(C_{ij})$ is the Cartan matrix of $\mathfrak{gl}_3$.
Therefore $\alpha_1=(1,-1,0)^T$ and $\alpha_2=(1,2,0)^T$. Then I obtain $\alpha_2 \neq e_2-e_3$. Where did I make mistake? Thank you very much.
 A: Weight theory only applies to semisimple Lie algebras. Now, $\mathfrak{gl}_3$ is reductive, and when people talk about weight theory they really mean weights for $\mathfrak{sl}_3$. 
Here is how I like to set things up in general: Let $\mathfrak{h}\subset\mathfrak{sl}_n$ be the standard Cartan subalgebra. The injective map
$$\mathfrak{h}\hookrightarrow\mathbb{C}^n,\;\;\;E_{ii}\mapsto e_i$$
induces a surjective map
$$(\mathbb{C}^n)^*\twoheadrightarrow\mathfrak{h}^*$$
with kernel spanned by $\epsilon_1+\cdots+\epsilon_n$ (here, $\epsilon_i$ is the coordinate function $\epsilon_i(e_j)=\delta_{ij}$). This yields an identification
$$\mathfrak{h}^*\cong(\mathbb{C}^n)^*/\langle \epsilon_1+\cdots+\epsilon_n\rangle.$$
Under this identification, the fundamental weights are
$$\omega_i=\epsilon_1+\cdots+\epsilon_i.$$ As cosets in the quotient space, the fundamental weights are also equal to $$\omega_i= (\epsilon_1+\cdots+\epsilon_i)-\frac{i}{n}(\epsilon_1+\cdots+\epsilon_n).$$
The fundamental roots are also only defined up to a shift by $\epsilon_1+\cdots+\epsilon_n$. In the specific example in your question,
$$
\alpha_2=(1,2,0)=(1,2,0)-(1,1,1)=(0,1,-1).
$$
