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

ok. so I know that gl(3) has basis \{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 V is a gl(3)-module if \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 \epsilon_1 - \epsilon_3 = (1, 0, -1, 0, 0, 0,...)

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