# Questions tagged [pfaffian]

Every question that is related to the Pfaffian polynomial and its traits.

13
questions

**9**

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**0**answers

226 views

### Local meaning of the Pfaffian of the curvature

The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can ...

**5**

votes

**1**answer

135 views

### Rational map given by pfaffians

Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...

**8**

votes

**2**answers

405 views

### Pfaffian representation of the Fermat quintic

It is known (see for instance Beauville - Determinantal hypersurfaces) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of ...

**12**

votes

**1**answer

616 views

### Determinant is to Pfaffian as resultant is to what?

This is an irresponsible question: I do not have done any thinking on it, or even literature search.
I just became curious whether there is some modification of the notion of a common root of two ...

**11**

votes

**1**answer

370 views

### Detailed modern references for basic properties of Pfaffians over commutative rings

Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing ...

**7**

votes

**2**answers

241 views

### Laplace-like / cofactor expansion for Pfaffian

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{\...

**5**

votes

**2**answers

226 views

### Pfaffian of several skew-linear transformations / matrices

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\...

**11**

votes

**1**answer

650 views

### Pfaffian equals complex determinant?

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure
$$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....

**3**

votes

**1**answer

257 views

### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...

**6**

votes

**2**answers

362 views

### Singularities of Pfaffian hypersurfaces

Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am ...

**31**

votes

**1**answer

2k views

### Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. ...

**2**

votes

**0**answers

157 views

### Orthogonality of Pfaffian polynomials in $SO(2m)$

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 = \...

**3**

votes

**3**answers

895 views

### Detecting if a polynomial is a Pfaffian

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?
The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ ...