MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.

Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = \left\{\oplus_{j=1}^m\begin{pmatrix} 0&v_j \\ -v_j&0\end{pmatrix} \big | v\in V\right\}$, with $v_j$ the $j$-th element of $v$, i.e., $S$ is the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$.

Now, let $I_m \in S$, $R\in SO(2m)$, and $g: \text{Skew}_{2m}\mapsto \mathbb{R}_+$ be a given function. I want to determine $\omega: \text{Skew}_{2m}\mapsto \mathbb{R}$, which is related to $g$ as follows: $$ g(R.I_m.R^T) = \int_{R^\prime \in SO(2m)} d \mu(R^\prime)\; \sum_{S_m\in S} \; \omega( {R^\prime}^T.S_m.{R^\prime})\; \text{Pf}({R^\prime}^T.S_m.{R^\prime}+{R}^T.I_m.{R}),$$ where $\mu$ is a uniform measure in $SO(2m)$, and $\text{Pf}$ is the Pfaffian.

One approach I've been trying, up to now wihtout success, is to use the integral form of Pfaffians: $$\text{Pf}(A) = \int d\theta_1\cdots \theta_{2m} e^{-\frac{1}{2} \theta^T.A.\theta},$$ where $\theta_i$, $i\in [2m]$, are Grassmann variables, i.e., $\theta_i \theta_j + \theta_j\theta_i=0$

Would you know how to tackle that? Any ideas?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.