**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!



Pfaffian of several skew-linear transformations / matrices
For two 2-forms I can offer you a blueprint of an algorithm. Details would have to be filled in.


Consider two skewsymmetric $n\times n$-matrices $A$ and $B$. The task is to compute $\operatorname{Pf}(sA+tB)$ since this gives all $\alpha^p\wedge\beta^{m-p}$ simultaneously.


Let's assume first that $A$ is invertible. Then there is a skew-analogue of Gram-Schmidt (first detail to be filled in) which brings $A$ into some sort of diagonal form $J$. Let $Sp(n)$ be the symplectic group with respect to $J$.


Now transform $B$ using a symplectic base change to bring it into an analogue of a companion matrix (second detail to be filled in). This is possible since the $Sp(n)$ action on skewsymmetric matrices is a $\theta$-representation in the sense of Vinberg.


Now $\text{Pf}(sA+tB)$ is readily computable.


If $A$ is not invertible, change it to $A+cB$ for a suitable constant $c$. If no such $c$ exists, the Pfaffian is identically zero. 


