Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\alpha\ (m\ {\rm times}) = n!\ ({\rm Pf}\ A)\cdot e_1\wedge\dots\wedge e_n$, where ${\rm Pf}\ A$ is the Pfaffian of the matrix $A$ whose elements are $a_{jk}$. In other words, if we need to calculate $(\alpha^{\wedge m},I)$, where $(\cdot,\cdot)$ is the scalar product and $I=e_1\wedge\dots\wedge e_n$, then we can easily do that: it is $n!\ ({\rm Pf}\ A)$, and the Pfaffian of a matrix $A$ is easily computed numerically, in polynomial time, for large $n$.
Question: Now, I have two 2-forms: one is $\alpha$ defined above, and the other is $\beta=(1/2)\sum_{j,k=1}^n b_{jk}\ e_j\wedge e_k$. I want to calculate $\alpha^{\wedge p}\wedge\beta^{\wedge m-p}$ for some $p$. What will be the coefficient in front of $I=e_1\wedge\dots\wedge e_n$? Can it be computed efficiently, say polynomially in $n$? In other words, how to numerically calculate $(\alpha^{\wedge p}\wedge\beta^{\wedge m-p},I)$ for some general skew-symmetric complex-valued matrices $A$ and $B$? What if I have not two, but $m$ 2-forms $\alpha_1,\dots,\alpha_m$, determined by the complex-valued matrices $A_1\dots A_m$, and we want to calculate ($\alpha_1\wedge\alpha_2\dots\wedge\alpha_m,I)$? Is it something easily computable, like Pfaffian of a matrix in the introduction above? In general, the answer is no, but what if the matrices $A_k$ are of low rank, say rank 4? (the case of rank 2 is trivial, since each form becomes a wedge product of two vectors)
Did someone study this, or saw some discussions about similar or related problems? Where can I try to find the information about such problems? I would appreciate any suggestion. I am not even sure what is the commonly used term for this coefficient.
What I found so far: in "Multilinear algebra" of W. Greub, the Pfaffian of skew-linear transformations is introduced (Ch. 8.4, Springer edition 1978) for several transformations. I.e. Greub starts as follows: $E$ is an inner product space of dimension $m=2n$, and $\phi_1, \phi_2,\dots \phi_m$ are the skew-linear transformations. Each $\phi_k$ determines an element $\Psi_k\in \bigwedge^2 E$, so that $\Psi_1\wedge\Psi_2\wedge\dots\Psi_m\in \bigwedge^n E$. Taking a basis vector $a\in E$, he defines ${\rm Pf}_a(\phi_1\dots\phi_m) = (\Psi_1\wedge\dots\wedge\Psi_m, a)$. This is exactly what I need. Unfortunately, already in the next paragraph he switches to the standard case $\phi_1=\phi_2=\dots=\phi_m=\phi$, which gives the standard answer for the Pfaffian of a matrix for $\Psi$, and does not discuss the case of different $\phi$'s anymore.
I would assume that people have studied this problem (or this is something simple, and I am just missing something obvious). I would be very grateful for help!
