Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states that det(partial) (det X)^s = s(s+1)...(s+n-1) (det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and partial=(partial/partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are the two-matrix and multi-matrix rectangular Cayley identities, the one-matrix rectangular antisymmetric Cayley identity, a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-parametrized" Cayley identities, and the "product-parametrized" and "border-parametrized" rectangular Cayley identities.