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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 against Wikipedia; it simply has different format than MO, and produces results of a different kind; both seem rather complementary), and in the spirit of the nice MO page Properties of functors and their adjoints, featuring compressed proofs or precise references to trustworthy references, of the known and not-overly-specializedexample 0 properties of Pfaffians.

A (rough) guideline could be to keep the list basic and context-free. (With 'context-free' roughly meaning: to keep to Pfaffians as an algebraic concept, and e.g. leave out any application to graphs without $K^{3,3}$-minors or even planar graphs).

A tentative list.

Definitions of the Pfaffian.

(0) The classical definition.editorializing 0

(1) The definition in Knuth: Overlapping Pfaffians.note on notation 0

(2) 'The' usual definition via exterior algebra.editorializing 1

(3) The definition of A. W. M. Dress, W. Wenzel: Advances in Mathematics Volume 112, Issue 1, April 1995, Pages 120-134.category-theoretic note 0

Basic properties of the Pfaffian.

Ideally, all the basic properties will be treated, with either a reasonably precise reference or a reasonably compressed proof. At the minimum, the properties listed---without satisfactory proof or reference--on the current Wikipedia page on Pfaffians will all be treated.

Some of these are the following.

(I suggest using indices in the finite ordinal $n$, in particular, the indexation starts at $0$.)

(0) For any commutative ring $R$ of characteristic different from two, and for each skew-symmetric $A\in R^{n\times n}$, if $n$ is odd, then $\mathrm{pf}(A)=0$.

(1) [to be filled with an appropriate property saying how Pfaffians behave under permutations of the factors of the cartesian-product-that-is-the-index-set; property (1) should be suitable to immediately deduce (2) from [Dress--Wenzel, (1.5), p. 122]; such a property is easy to come by; the touchy point is whether to make (1) into more than a mere instrument to prove (2) ]

(2) For any commutative ring $R$, and for any $i\in n$, and with $[\cdot]$ being the Iverson bracket and with $|_{}$ being the usual restriction notation applied to matrices-construed-as-functions $n\times n\to R$, for any skew-symmetric $A\in R^{n\times n}$,

$\mathrm{pf}(A)=\sum_{j\in n\setminus\{i\}}\ (-1)^{\bigl[j<i\bigr]+j+i}\cdot A|_{(i,j)}\cdot\mathrm{pf}\bigl( A|_{ (n\setminus\{i,j\})\times(n\setminus\{i,j\}) } \bigr)\ .$

(3) For any commutative ring $R$, and for even $n$, and for any skew-symmetric $A\in R^{n\times n}$,

$\mathrm{pf}(A^{\mathrm{t}})=(-1)^{n/2}\cdot \mathrm{pf}(A)$.

(4) For any commutative ring $R$, any $\lambda\in R$ and for any skew-symmetric $A\in R^{n\times n}$,

$\mathrm{pf}(\lambda\cdot A)=\lambda^{n/2}\cdot \mathrm{pf}(A)$.

(5) For any commutative ring $R$, and arbitary skew-symmetric $A,B\in R^{n\times n}$,

$\mathrm{pf}(BAB^{\mathrm{t}})=\det(B)\cdot\mathrm{pf}(A)$.

(6) For any commutative ring $R$, and arbitary finite square skew-symmetric matrices $A$ and $B$ with entries from $R$,

$\mathrm{pf}\biggl( \array{A & 0 \\ 0 &B}\biggr) = \mathrm{pf}(A)\cdot\mathrm{pf}(B)$

(7) For any commutative ring $R$, and for any $A\in R^{n\times n}$,

$\mathrm{pf}\biggl( \array{0 & A \\ - A^{\mathrm{t}} &0}\biggr) = (-1)^{\frac12 n(n-1)}\cdot \det(A)$

(8) For any integraleditorializing 3 domain $R$, and for any skew-symmetric $A\in R[x]^{n\times n}$, calculating in the field of fractions of $R[x]$ we have

$\frac{1}{\mathrm{pf}(A)}\cdot \frac{\mathrm{d}}{\mathrm{d} x} \mathrm{pf}(A) = \frac12\cdot \mathrm{tr}\bigl( A^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d} x} A \bigr) $

(9) For any commutative ring $R$, and arbitary skew-symmetric $A,B\in R^{n\times n}$,

$\mathrm{pf}(A)\cdot\mathrm{pf}(B) = \exp(\frac12\cdot\mathrm{tr}\ \log (A^{\mathrm{t}} B))$

(If you think more entries are appropriate here, or please add some. Of course, 1. is a trivial application of the morphism $\mathrm{Mat}(n\times n;R)\xrightarrow[]{\mathrm{det}}R$. It is rather the properties starting from 2. which this thread will hopefully be useful for. In particular, is there a citable source for the property 2., in essentially this form, which relieves one of the responsibility of explicitly doing row-and-colum-permutations to get make the $i$-'normalized'-to-1-version-of-the-formula in e.g. [Dress--Wenzel, p. 122, (1.5)] and in Godsil available? This is important when doing explicit hand-calculations with moderately large explicit matrices over a polynomial ring, since for the 'normalized version' one would have to re-scribble the entire matrix again only to do the permutations, while with the general formula 2. one can continue to 'operate' 'in-place'. The Wikipedia reference does not lead anywhere. Of course one could work this out for oneself, taking Godsil's nice exposition as the roadmap.)

Proofs of the basic properties.

Ad 0: This needs no documenting here. In characteristic other than two this can be found in many places. In characteristic two this would probably turn into something unusual and context-dependent.

Ad 1: Easy to see using the sign-homomorphism $\mathrm{sgn}$ and the 'classical definition'.

Ad 2: This follows immediately from [Dress--Wenzel, p. 122, (1.5)], and [a still-to-be-written appropriate entry in] (1).

Commentary.

Part of the motivation is that it seems to me that, while there surely is more than enough documentation on this somewhere, it is currently unsatisfactorily accessible.

In particular, the current Wikipedia page, while containing proofs and references here and there, currently makes no attempt to give references for the non-trivial entries in its section 'Properties and identities'. I recently had to do explicit calculations with Pfaffians, involving operations like 'simultaneously permuting rows and colums' (to respect the skew-symmetry) and 'recursive expansion by deleting two rows and colums simultaneously' (which is partially treated in Godsil's nice monograph cited below, 'partially' because there, one of the indexes is fixed to be '1', while e.g. the Wikipedia page claims without proof the more flexible formula,

enter image description here

and 'partially' also because there the actual formula for row-expansion was relegated to an exercise:

enter image description here )

and was much slowed down by the necessity to make sure that even the basic tools like the general expansion formula (2. below) are used correctly.

(In my case, for context-dependent reasons irrelevant to this thread, the flexibility of having the arbitrary $i\in n$ in Property 2, made much of a difference, and could hardly be simulated by the formula in Godsil's monograph.)

References

Some useful references are:

  • The only rigorous and modern book treatment known to me is

C. D. Godsil: Algebraic Combinatorics, Chapman and Hall 1993.

Then are also:

There are also

  • a monograph of Muir whose preface is dated 1905,

  • the current Wikipedia page on Pfaffians, which leaves much to be desired: there are malapropisms (as far as I can tell, what Wikipedia calls a 'gradient' is just a formal derivative), there are broken links (of all things, precisely the very link ${}^{\color{blue}{\text{$[2]$}}}$ which purports to be a reference for the more flexible expansion-formula cited above is broken), there are missing references, there are sketchy proofs and there are inconclusive arguments on the talk-page); my hope is that the properties listed here will eventually be more precise and usable and kept up to date by watchful eyes.

  • C. E. Cullis: Matrices and Determinoids. Cambridge University Press 1918, is very relevant, in particular explicitly addresses expansion of Pfaffians by rows and columns, in

enter image description here

yet this book is a century old and, at times, to me, reads like a parody, starting with 'determinoid' in its title. (I have no reasons to doubt its correctness, and expect that a modicum of tolerance towards terms like 'the affect of $[i,j]$' will probably make it perfectly usable.)

One could also try to find out whether there exists a good category with which to compare all the existing definitions of the Pfaffian.

category-theoretic note 0 Dress and Wenzel's definition is reminiscent of a definition by a universal mapping property, yet strictly speaking it is not a universal mapping property in the usual sense: a pure universal mapping property, roughly speaking, is a formula, starting with universal quantifiers (whose range is a subtle matter touching on set-theoretic issues), over the signature of a category. (Possibly a higher category). Dress and Wenzel's interesting definition avails oneself of a larger signature: the multiplication in the ground-ring still figures in the definition. Also: there are remnants of arithmetic in their definition. Also: it is not clear what the (right) category to use with loc. cit. definition is. One way to put it would be to say that loc. cit. works in the category of categories, characterizing the Pfaffian as a 2-cell of $\mathsf{Cat}$ with the characteristic property of satisfying a univerally quantified formula over a larger signature than $\mathsf{Cat}$ natively has. (Note that currently this is little more than a restatement of the definition of Dress and Wenzel, replacing the term 'natural transformation' by '2-cell in $\mathsf{Cat}$.) I do not know whether one can say anything useful 2-categorically about Dress and Wenzel's definition in the 2-category $\mathsf{Cat}$. In particular I do not know whether there are precedents in the literature for trying to find a definition by a universal property in the 2-category $\mathsf{Cat}$ of a 2-cell $P$ of $\mathsf{Cat}$ when $P$ was defined by a universally quantified formula over the signature of the theory of commutative rings (and a little index-arithmetic).

editorializing 0 Saying much about this perfectly fine definition would perhaps be distracting and superfluous. One could perhaps say something useful about whether one should extend the classical definition to non-skew-symmetric matrices (losing some well-definedness), but perhaps one shouldn't tamper with the definition at all.

editorializing 1 There are some subtleties in this, and moreover the use of exterior algebra tends to, yet not necessarily has to, bring context-dependent interpretations in its wake, such as in the (by itself very interesting) Does the Pfaffian have a geometric meaning?. There is also the problem of giving precise references to the literature when it comes to exterior algebras of free modules over integral domains, which seem still underpresented in the book literature, where there is a preponderance of exterior algebras of vector spaces .

editorializing 3 Then $R[x]$ is an integral domain, too, hence has a field of fractions, making the $\frac{1}{\mathrm{pf}(A)}$ in the formula defined. Moreover, the use of fractions here seems an unnecessary complication in the Wikipedia treatment, and should be possible to prove this for any commutative ring, written $\frac{\mathrm{d}}{\mathrm{d} x} \mathrm{pf}(A) = \frac12\cdot \mathrm{pf}(A)\cdot \mathrm{tr}\bigl( A^{-1} \cdot \frac{\mathrm{d}}{\mathrm{d} x} A \bigr)$.

example 0 The research literature e.g. contains quite a bit on Pfaffians of special families of 'structured matrices'. This is fine yet perhaps too specialized for this thread. It seems to me (I might of course be overlooking something) that even something as basic as the recursive formula by 'expansion along a row' does not have any easily accessible reference anywhere. Yes, Godsil's monographs offers all what is necessary to rigorously prove it, yet the formula is spread out between the main text and an exercise.

note on notation 0 Knuth's is essentially the classical definition. A dictionary 'Knuth : here' is given by '$f$ : $A$' , '$x$ : $i$' , '$y$ : $j$' , '$f[x_1\dotsc x_{2n}]$ : $A$', '$2n$ : $n$' ; note Knuth uses '$x$' simply as an indexing-variable; Knuth's $f[x_1\dotsc x_n]$ is not a polynomial in the $x_i$, despite a superficial resemblance; one may interpret his choice of square brackets as a way to eschew an even stronger resemblance to the usual polynomial-notation '$f(x_1,\dotsc,x_n)$'; Knuth's Pfaffians are polynomials in the $f[xy]$, i.e., for Knuth, Pfaffians are elements of $\mathbb{Z}\bigl[ f[xy]\colon (x,y)\in\{1,\dotsc,n\}^2\bigr]$, with '$f[xy]$' being a formal term having three free variables '$f$', '$x$', '$y$', with only the latter two being usually used.

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    $\begingroup$ I would suggest adding Knuth's article Overlapping Pfaffians to your list of references and properties. Perhaps much of it concerns identities for "structured" matrices, but they are as fundamental as the Vandermonde is for the determinant, so I don't think they are over-specialized. $\endgroup$ – Dave Anderson Sep 5 '17 at 1:54
  • $\begingroup$ @DaveAnderson: I added three references, none of them overly specialized, one of them the one you suggested. $\endgroup$ – Peter Heinig Sep 5 '17 at 6:04
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    $\begingroup$ +1! I was starting to write a detailed expository about Pfaffians last term, which would lead up to a proof of the minor summation formula, but I got pretty much nowhere until time constraints kicked it way down my to-do list. Let me add Nicholas Loehr, Bijective Combinatorics as a textbook that does at least the basics of the theory rather cleanly. $\endgroup$ – darij grinberg Sep 5 '17 at 11:42
  • $\begingroup$ @darijgrinberg: thanks for these comments. Feel free to edit at your convenience. For today, I will have to leave this thread. Thanks again, and let's hope this thread will evolve into something useful. $\endgroup$ – Peter Heinig Sep 5 '17 at 11:46
  • $\begingroup$ In (8), you might be able to get rid of the integral-domain condition by replacing the inverse $A^{-1}$ by the adjoint or the "Pfaffian adjoint". What do I mean by "Pfaffian adjoint"? See dropbox.com/s/utyaljkl1dsddwl/todo1205.pdf?dl=0 (which is also a little expository to-do list of mine). $\endgroup$ – darij grinberg Sep 5 '17 at 11:48
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The Pfaffian satisfies an analogue of the Matrix-Tree Theorem. This was discovered by Masbaum and Vaintrob. I also found a generalization to Hyperpfaffians.

One should note that Pfaffians are the same as Berezin integrals which give a more powerful point of view than just "(2) 'The' usual definition via exterior algebra" as can be seen from the availability of nontrivial tools such as Berezin's supersymmetric change of variable formula.


As per Darij's question: This is a generalization of the change of variable formula in multiple integrals with the Berezinian playing the role of the Jacobian. For a presentation suitable for analysts see Appendix A of the article "Supersymmetric measures and maximum principles in the complex domain. Exponential decay of Green's functions" by Sjöstrand and Wang. For a presentation more suitable to geometers and topologists see Section 3.10 of the "Notes on Supersymmetry (following Joseph Bernstein)" by Deligne and Morgan in Quantum fields and strings: a course for mathematicians, Vol. 1.

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    $\begingroup$ "Berezin's supersymmetric change of variable formula" what is that? $\endgroup$ – darij grinberg Sep 5 '17 at 15:38
  • $\begingroup$ Thanks for the elaborations; but I'm still lost as to where the Pfaffian appears. I always thought (maybe incorrectly!) that the Pfaffian is a type-B analogue of the determinant, not a super-analogue of it! $\endgroup$ – darij grinberg Sep 7 '17 at 17:07
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    $\begingroup$ Ah, the Pfaffian appears as a Grassmann-Berezin integral in Proposition A.13 of arxiv.org/abs/1105.6270v2 ; this explains what you mean! $\endgroup$ – darij grinberg Sep 7 '17 at 17:09

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