Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like $$ u = \sum_{r \geq \max(k-n,0)} L^r u_r $$ for some $(k-2r)$-forms $u_r$ such that $\Lambda u = 0$. This is proved in many places, but there are not many examples or calculations that I've seen!
For example, if I write down some form $\eta$ on $\mathbb{C}^n$ (which is a Kahler manifold when equipped with the Kahler form $\omega = \frac{i}{2} \sum_{j=1}^n dz_j \wedge d\overline{z}_j$), then how would I find the primitive decomposition of $\eta$?
Is there some algorithmic way, in general, to calculate the primitive decomposition of a form?
The decomposition can be calculated recursively. Kind of.
Let's consider $(p,q)$-classes with $k := p + q \leq n$ (because for $k > n$ there are no primitive classes) and the operator $L\Lambda$. It is an endomorphism of the space of $(p,q)$-classes and has a characteristic polynomial $p(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_1 x + c_0$. Since primitive classes exist we have $c_0 = \det (L\Lambda) = 0$. By the Cayley-Hamilton theorem we then have $$ 0 = p(L\Lambda) = (L\Lambda)^n + c_{n-1}(L\Lambda)^{n-1} + \cdots + c_1 L\Lambda = (L\Lambda)^m f, $$ where $f := (L\Lambda)^{n-m} + c_{n-1} (L\Lambda)^{n-m-1} + \cdots + c_m \operatorname{id}$ and $c_m \not= 0$.
We then have $(L\Lambda)^m f(u) = 0$ for every class $u$. As $L$ is injective, we conclude that $(L\Lambda)^{m-1} f(u)$ is primitive. But $Lv$ is never primitive for a nonzero class $v$, so by induction on $m$ we must in fact have that $f(u)$ is primitive.
Now suppose $u$ is primitive. Then $f(u) = c_m u$, so by rescaling $f$ we obtain a projection operator onto the space of primitive classes. Then the Hodge theorems tell us that $$ u - \frac{1}{c_m} f(u) = Lv $$ for some $(p-1,q-1)$-class $v$, and we recurse.
This is simplest for a $(1,1)$-class $u$, where we get that $$ u = \Bigl(u - \frac1n L\Lambda u\Bigr) + \frac1n L\Lambda u. $$ You can work out what this gives for $(2,2)$-classes as well, but it's not fun.