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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.

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  • $\begingroup$ Where does that relation between simple roots and fundamental weights come from? $\endgroup$ – Tobias Kildetoft Sep 11 '18 at 12:42
  • $\begingroup$ @TobiasKildetoft, thank you very much. I edited the post. It comes from the Cartan matrix. $\endgroup$ – LJR Sep 11 '18 at 12:45
  • $\begingroup$ Ahh, it should actually be clear that those expressions for the fundamental weights could not be right with those choices of the roots, as all linear combinations of the fundamental weights would have the last component equal to $0$ (so while one can probably make them have that form, it is not possible while also having the roots of the given form). $\endgroup$ – Tobias Kildetoft Sep 11 '18 at 12:54
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    $\begingroup$ If you think of $\mathfrak{sl}_3$ instead, everything is fine since $(1,1,1)$ should be identified with $0$. $\endgroup$ – Ennar Sep 11 '18 at 12:57
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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, $$ \alpha_2=(1,2,0)=(1,2,0)-(1,1,1)=(0,1,-1). $$

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