nLab Pfaffian




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


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


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

(1)Pf(A)12 nn! σSym 2nsgn(σ) i=1 nA σ(2i1),σ(2i), 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)} \,,


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

(2)Pf(A)12 nn!A i 1j 1A i 2j 2A i nj nϵ i 1j 1i 2j 2i nj n. 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 } \,.


Relation to the determining


(Pfaffian is square root of determinant)

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

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

(3)(Pf(A)) 2=det(A). \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


Let Λ 2n\Lambda_{2n} be the Grassmann algebra on 2n2n generators {θ i}\{\theta_i\}, which we think of as a vector θ\vec \theta

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

Pf(A)=exp(θ,Aθ)dθ 1dθ 2dθ 2n. Pf(A) = \int \exp( \langle \vec \theta, A \cdot \vec \theta \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} \,.

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

det(A)exp(θ,Aψ)dθ 1dθ 2dθ 2ndψ 1dψ 2dψ 2n. 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 NN spinless electrons

Ψ Pf(z 1,,z N)=pfaff(1z kz l) i<j(z iz j) qexp(14|z| 2) \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 kl)pfaff(M_{k l}) denotes the Pfaffian of the matrix whose labels are k,lk,l and q=1/νq= 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




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

Euler forms

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

Last revised on March 17, 2020 at 00:49:10. See the history of this page for a list of all contributions to it.