## orthonormal basis

give an orthonormal basis for the $\displaystyle gl(3)$-module $\displaystyle V(\epsilon_1 - \epsilon_3)$. give the matrix of the generators $\displaystyle a^1_2, a^2_1, a^1_1$ and $\displaystyle a^2_2$ in the corresponding representation, and give the branching rule for $\displaystyle gl(2) \subset gl(3)$ in this case.

ok. so I know that $\displaystyle gl(3)$ has basis $\displaystyle \{a^1_1, a^1_2, a^1_3, a^2_1, a^2_2, a^2_3, a^3_1, a^3_2, a^3_3\}$, that $\displaystyle V$ is a $\displaystyle gl(3)$-module if $\displaystyle \forall \alpha, \beta \in \mathbb{C}, v, w \in V, x_1, x_2 \in gl(3)$ linearity and bracket preservation hold, and I know that $\displaystyle \epsilon_1 - \epsilon_3 = (1, 0, -1, 0, 0, 0,...)$

because the basis of $\displaystyle gl(3)$ has 9 elements, does that mean that the orthonormal basis for the $\displaystyle gl(3)$-module $\displaystyle V(\epsilon_1 - \epsilon_3)$ will also have 9 elements? and does $\displaystyle \epsilon_1 - \epsilon_3 = (1, 0, -1, 0, 0, 0, 0, 0, 0)$? im also a little confused as to what the $\displaystyle gl(3)$-module $\displaystyle V(\epsilon_1 - \epsilon_3)$ actually is...any help AT ALL will be greatly appreciated!