Highest or positive weights (or roots) Let $T= (\mathbb{C}^*)^2$ be embedded in $GL_2$ along its diagonal entries, and suppose $T$ acts on $M_2(\mathbb{C})$ via conjugation. Denote $\chi_i(g)=z_i$ where 
$$
g = \left( \begin{array}{cc}
z_1 & 0 \\
0 & z_2 \\ 
\end{array}\right). 
$$
Then we have a weight space decomposition of $M_2(\mathbb{C})$: 
$$
M_2(\mathbb{C}) = \underbrace{\mathbb{C}\cdot E_{12}}_{\mbox{weight } \chi_1\chi_2^{-1} } \oplus \underbrace{\mathbb{C}\cdot E_{21}}_{\mbox{weight }\chi_1^{-1}\chi_2} \oplus \underbrace{  \left\{ \left( 
\begin{array}{cc}
a & 0 \\ 
0 & b \\ 
\end{array}
\right) : a, b \in \mathbb{C}\right\} }_{\mbox{weight } 0}
$$
where $E_{ij}$ is in $M_2(\mathbb{C})$ with a 1 in the $(i,j)$-entry and zeros everywhere else. 

How does one determine which one is the highest or positive weight? Or it's possible that I am misunderstanding weights with roots; are they related? 

When I mean roots, I am thinking of those that arise in root systems. 
Added: please feel free to impose any noncanonical ordering on the above weights if there doesn't exist a canonical choice. 

Added question: under what conditions or circumstances would there be a canonical choice on the weights so that the highest weight can be determined? Any simple example would be fine. 

Thank you. 
 A: You have a Lie group $G=GL(2,\mathbb C)$ with a chosen maximal torus $T$, acting on its Lie algebra $M_2(\mathbb C)$ by conjugation (the adjoint representation of $G$), so the roots are precisely the non-zero weights $\alpha=\chi_1\chi_2^{-1}$ and $\beta=\chi_1^{-1}\chi_2$ (considered as characters of $T$, that is algebraic group morphisms $T\to\mathbb C^\times$). Note that since the group $\mathrm{Hom}(T,\mathbb C^\times)$ of all characters is isomorphic to $\mathbb Z^n$ (here $n=2$), it is customary to write characters additively rather than multiplicatively: $\beta=-\alpha$ rather than $\beta=\alpha^{-1}$, and indeed $\alpha= \chi_1-\chi_2$ rather than $\alpha=\chi_1\chi_2^{-1}$.
In order to give sense to the notions of "positive root" and that of "highest weight" derived from it, one needs a choice between a finite number (here $2$) of possibilities, conveniently embodied by the choice of a Borel subgroup $B$ of $G$ containing $T$. Here the two possible choices are that of the group of upper triangular matrices in $G$ and the group of lower triangular matrices; it is conventional to choose the upper ones. Once that is done, the roots whose root subspace lies in the Lie algebra of $B$ are called positive, and the others negative. Here the root subspace $E_{1,2}$ lie in that Lie algebra, so $\alpha$ is considered positive, and $\beta=-\alpha$ negative. Thereby $\alpha$ is also the highest weight of the adjoint representation (but the adjoint representation is not always a highest weight representation). In general a weight of a representation is a highest weight if all other weights can be obtained from it by subtracting (a multiset of) positive roots.
