Overlapping Pfaffians

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. 0. Definitions. Let X be a possibly infinite index set. We consider quantities f [xy] defined on ordered pairs of elements of X , satisfying the law of skew symmetry f [xy] = −f [yx] , for x, y ∈ X . (0.0) This notation is extended to f [α] for arbitrary words α = x1 . . . x2n of even length over X by defining the Pfaffian f [x1 . . . x2n] = ∑ s(x1 . . . x2n, y1 . . . y2n) f [y1y2] . . . f [y2n−1y2n] , (0.1) where the sum is over all (2n − 1)(2n − 3) . . . (1) ways to write {x1, . . . x2n} as a union of pairs {y1, y2} ∪ · · · ∪ {y2n−1, y2n}, and where s(x1 . . . x2n, y1 . . . y2n) is the sign of the permutation that takes x1 . . . x2n into y1 . . . y2n. The Pfaffian is well defined, even though there are 2n! different permutations y1 . . . y2n that yield the same partition {y1, y2}∪ . . .∪{y2n−1, y2n} into pairs. For if we interchange y2j−1 with y2j , we change the sign of both s(x1 . . . x2n, y1 . . . y2n) and f [y1, y2] . . . f [y2n−1yn], by (0.0); if we interchange y2i−1 with y2j−1 and y2i with y2j , both factors stay the same. Thus, for example, f [wxyz] = f [wx]f [yz]− f [wy]f [xz] + f [wz]f [xy] = f [wx]f [yz] + f [wy]f [zx] + f [wz]f [xy] . (0.2) A partition into pairs is commonly called a perfect matching. Therefore it is convenient to abbreviate (0.1) in the form f [α] = ∑ μ∈M(α) s(α,μ) Πf [μ] (0.3) where M(α) is the set of perfect matchings of α represented as words y1 . . . y2n in some canonical way, and Πf [y1 . . . y2n] = f [y1y2] . . . f [y2n−1y2n]. Notice that we have f [wxyz] = −f [xyzw] . (0.4) In general, an odd permutation of α will reverse the sign of f [α], because every term in (0.3) changes sign. 1 Pfaffians can also be defined recursively, starting with the null word 2 and proceeding to words of greater length: f [2] = 1 ; f [x1 . . . x2n] = 2n ∑ j=2 f [x1xj ]f [xj+1 . . . x2nx2 . . . xj−1] , n > 0 . (0.5) This recurrence [9] corresponds to a procedure that constructs all perfect matchings by starting with {x1, x2}∪· · ·∪{x2n−1, x2n} and making cyclic permutations of the indices in positions {2, . . . , 2n}, {4, . . . , 2n}, . . . ; each of these permutations is even. It will be convenient in the sequel to extend the sign function s to s(α, β) for arbitrary words α, β ∈ X∗. We define s(α, β) = 0 if either α or β has a repeated letter, or if β contains a letter not in α. Otherwise s(α, β) is the sign of the permutation that takes α into the word β (α\β) , where α\β is the word that remains when the elements of β are removed from α. Thus, for example, s(αβγ, β) = {0 , if αβγ contains a repeated letter; (−1)|α| |β| , otherwise. (0.6) We also have s(α, βγ) = s(α, β)s(α\β, γ) , (0.7) since both sides vanish unless the letters of βγ are distinct and contained in the distinct letters of α, and in the latter case s(α, βγ) is the parity of the number of transpositions needed to bring β to the left of α and γ to the left of the remaining word α\β. If α has repeated letters, the Pfaffian f [α] is zero, because f [α] = −f [α] when we transpose two identical letters. Therefore our convention that s(α, β) = 0 when α or β has repeated letters does not invalidate definition (0.1), which used a different convention for s(x1 . . . x2n, y1 . . . y2n). One consequence of the new convention is the identity f [α] = ∑ x1<···<xn ∑