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.