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!


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.

  • $\begingroup$ First of all, thank you! The approach you suggested looks very interesting and thought-provoking; maybe it can even be extended to the case of many matrices $A_1\dots A_m$. Now, about the details to be filled in: the first one I clear, it is not difficult to do. But I am a little puzzled by the second one, and I would greatly appreciate if you could elucidate it a little. What do you mean by an "analogue of the companion matrix"? Companion matrix to which polynomial? Did you mean just the convenient form that the companion matrix has, or do I miss something in your answer? $\endgroup$ – BarTov May 16 '16 at 3:22
  • $\begingroup$ What I meant was "rational canonical form" aka. "Frobenius normal form". The polynomial $\text{Pf}(sJ+B)$ is a skew-symmetric analogue of the characteristic polynomial. Maybe we don't have to reinvent the wheel.I googled for "ratonal canonical form" and "skew-symmetric matrix" and stumbled across the paper "Skew-Symmetric Matrix Polynomials and their Smith Forms" by Mackey, Mackey, Mehly, and Mehrmann. The proof of Thm. 5.4 seems to give the algorithm you are looking for. $\endgroup$ – Friedrich Knop May 16 '16 at 6:45
  • $\begingroup$ Finally, no, I don't think that this generalizes to more than two $2$-forms. The invariant theory of $GL(n)$ acting on three or more copies of $\wedge^2\mathbb{C}^n$ is too bad. $\endgroup$ – Friedrich Knop May 16 '16 at 6:49

Since the two forms in the definition of "multilinear Pfaffian" commute one can use polarization and write down explicit formula. See e.g. (7) of https://arxiv.org/abs/1309.1275 This formula, however, has $2^m$ terms.


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