nLab Pfaffian

Idea

The Pfaffian of a skew-symmetric matrix is a square root of its determinant.

Definition

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-matrix with entries in some field (or ring) $k$.

Definition

The Pfaffian $Pf(A) \in k$ is the element

(1)$$Pf(A) \;\coloneqq\; \frac{1}{2^n n!} \sum_{\sigma \in Sym_{2n}} sgn(\sigma) \prod_{i = 1}^n A_{\sigma(2i -1), \sigma(2i)} \,,$$

where

• $\sigma$ runs over all permutations of $2n$ elements;

• $sgn(\sigma)$ is the signature of a permutation.

Expressed equivalently in terms of the Levi-Civita symbol $\epsilon$ and using the Einstein summation convention the Pfaffian is

(2)$$Pf(A) \;\coloneqq\; \frac{1}{2^n n!} A_{i_1 j_1} A_{i_2 j_2} \cdots A_{i_n j_n} \epsilon^{ i_1 j_1 i_2 j_2 \cdots i_n j_n } \,.$$

Properties

Relation to the determining

Proposition

(Pfaffian is square root of determinant)

Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-matrix with entries in some field (or ring) $k$.

Then the Pfaffian of $A$ (1) is a square root of the determinant of $A$:

(3)$$\big( Pf(A) \big)^2 \;=\; det(A) \,.$$

Proofs are spelled out for instance in Haber 15, Sections 2 and 3

In terms of Berezinian integrals

Proposition

Let $\Lambda_{2n}$ be the Grassmann algebra on $2n$ generators $\{\theta_i\}$, which we think of as a vector $\vec \theta$

Then the Pfaffian $Pf(A)$ is the Berezinian integral

$$Pf(A) = \int \exp( \langle \vec \theta, A \cdot \vec \theta \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} \,.$$

Remark

Compare this to the Berezinian integral representation of the determinant, which is

$$det(A) \propto \int \exp( \langle \vec \theta, A \cdot \vec \psi \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} d \psi_1 d \psi_2 \cdots d \psi_{2n} \,.$$

Pfaffian state

Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of $N$ spinless electrons

$$\Psi_{Pf}(z_1,\ldots,z_N) = pfaff\left(\frac{1}{z_k-z_l}\right)\prod_{i\lt j}(z_i-z_j)^q exp(-\frac{1}{4}\sum |z|^2)$$

where $pfaff(M_{k l})$ denotes the Pfaffian of the matrix whose labels are $k,l$ and $q= 1/\nu$ is the filling fraction, which is an even integer. For Pfaffian state see

• Gregory Moore, N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. 360B(1991)362 pdf

Related entries

• determinant,

• Pfaffian line bundle,

• hafnian

References

General

Basics:

• Howard E. Haber, Notes on antisymmetric matrices and the pfaffian, 2015 (pdf, pdf)

See also

• J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026

• Claudiu Raicu, Jerzy Weyman, Local cohomology with support in ideals of symmetric minors and Pfaffians, arxiv/1509.03954

• Haber, Notes on antisymmetric matrices and the pfaffian, pdf

There is also a deformed noncommutative version of Pfaffian related to quantum linear groups:

• Naihuan Jing, Jian Zhang, Quantum Pfaffians and hyper-Pfaffians, Adv. Math. 265 (2014), 336–361, arxiv/1309.5530

Pfaffian variety is subject of 4.4 in

• Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, arxiv/1404.3143

Relation to $\tau$-functions is discussed in

• J. W. van de Leur, A. Yu. Orlov, Pfaffian and determinantal tau functions I, arxiv/1404.6076

Other articles:

• András C. Lőrincz, Claudiu Raicu, Uli Walther, Jerzy Weyman, Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians, arxiv/1601.06688

• Mauro Spera, A C–algebraic approach to determinants and Pfaffians, Acta Cosmologica Fasc. XXI-2, (1995) 203–208.

• Donald E. Knuth, Overlapping Pfaffians, Electronic Journal of Combinatorics 3:2 (1996) pdf

  “A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.”

• Jacques Distler, Nathan Donagi, Ron Donagi: On Generalized Pfaffians [arXiv:2409.06871]

Euler forms

Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

• Shiing-Shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annals of Mathematics Second Series 45:4 (1944) 747-752 (jstor:1969302)

• Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

• Siye Wu, Section 2.2 of Mathai-Quillen formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

• Gerard Walschap, ch. 6.3 of Metric structures in differential deometry, Graduate Texts in Mathematics, Springer 2004

• Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)

• Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)

category: algebraic geometry, representation theory