Laplace-like / cofactor expansion for Pfaffian

Question

Wikipedia presents a recursive definition of the Pfaffian of a skew-symmetric matrix as $$ \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_{ij}\operatorname{pf}(A_{\hat{\imath}\hat{\jmath}}),$$ where $\theta$ is the Heaviside step function. I have failed to find a proper reference for this result. Is it standard or does it require proper citation? What should one cite?

Answer 1

Your formula can be found in

Fulton, William, Pragacz, Piotr: Schubert varieties and degeneracy loci, Lecture Notes in Mathematics 1689, Springer. xi, 148 p. (1998). ZBL0913.14016,

see in particular equation (D.1) p. 116.

Answer 2

Here is how you get the Pfaffian. $\newcommand{\bR}{\mathbb{R}}$ Consider the symplectic $2$-form

$$ \omega = dx^1\wedge dx^2+\cdots +x^{2n-1}\wedge x^{2n} \in\Omega^2(\bR^{2n}). $$

Note that

$$ \frac{1}{n!}\omega^{\wedge n}:=\frac{1}{n!}\underbrace{\omega\wedge \cdots \wedge \omega}_n=dx^1\wedge dx^2\wedge \cdots \wedge dx^{2n-1}\wedge dx^{2n} $$

Suppose that $A=(a_{ij})$ is a skew-symmetric $2n\times 2n$ matrix. None that $\newcommand{\be}{\boldsymbol{e}}$

$$a_{ij}=(\be_i, A\be_j), $$

where $(\be_i)$ is the canonical basis of $\bR^{2n}$. Consider now the $2$-form

$$ \omega_A=-\sum_{i<j} aa_{ij}dx^i\wedge dx^j=\sum_{i<j} (A\be_i,\be_j) dx^i\wedge dx^j=-\frac{1}{2}\sum_{i\neq j} a_{ij} dx^i\wedge dx^j. $$

The top exterior power of $\omega_A$ is a multiple $\mu$ of $\frac{1}{n!}\omega^{\wedge n}$

$$\omega_A^{\wedge n}=\frac{\mu}{n!} \omega^{\wedge n}. \tag{*}$$

The Pfaffian of $A$ is then the multiple $\mu$. If you write $\omega_A$ in the form

$$\omega_A =-\frac{1}{2}\sum_j a_{ij} dx^i\wedge dx^j-\frac{1}{2}\sum_{k,\ell\neq i} a_{k\ell} dx^k\wedge dx^\ell $$ and use the equality (*) then you will get your formula.