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

摘要

The classic Cayley identity states thatdet(?)(~(detX))s=s(s+1) ...(s+n-1)(~(detX)s-1) where X=(~(xij)) is an n×n matrix of indeterminates and ?=(?/?~(xij)) is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/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 a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-parametrized" Cayley identities, and the "product-parametrized" and "border-parametrized" rectangular Cayley identities.