quantum mechanics - Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons.

I've got the following question to work out;

Let $F_2$ denote the Fock subspace of all 2-boson states, spanned by homogeneous polynomials of degree 2 in the boson creation operators $b^{\dagger}_1$, $b^{\dagger}_{2}$, $b^{\dagger}_{3}$ acting on the physical vacuum state $\left|0\right\rangle$. From lectures, we know that $F_2$ is a six dimensional $gl(3)$-module. Explicitly give an orthonomal basis of $F_2$ that is symmetry adapted to the subalgebra chain $o(2) \subset o(3) \subset gl(3)$, and express the highest weight vector of $F_2$ as a linear combination of these basis vectors.

Now, just going through lecture notes, I tried to work out a few things step by step.

• The $gl(3)$ highest weight vector of $F_2$ must be $(2, 0, 0)$


• $F_2 = c_2 \dotplus \Delta^{\dagger}F_0$


• dim($c_2$) = 5


• $c_2 = \{\Psi \in F_2 | \Delta\Psi = 0\}$


• Define $\Psi_2 = k(b^{\dagger}_1 + i b^{\dagger}_2)^2 \left|0\right\rangle$

Now, I'm not even sure where to go from there. I've been told to normalise that $\Psi_2$ term, but I'm not even sure how to go about that. I've also been told to act $L_3$ on the $\Psi$ terms, where

$$L_3 = -i(b^{\dagger}_{1}b_{2} - b^{\dagger}_{2}b_{1})$$

But I'm not even sure how to even start going about this. Of course I'd love some help on this question, but, links to easy-ish resources that might point me in the right direction, or even give some better background into this would also be much appreciated.