Combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
نویسندگان
چکیده
The classic Cayley identity states that det(∂) (detX) = s(s+ 1) · · · (s+ n− 1) (detX) 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 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 “diagonalparametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities. Also at Department of Mathematics, University College London, London WC1E 6BT, England.
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