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

Abstract

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 a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-parametrized" Cayley identities, and the "product-parametrized" and "border-parametrized" rectangular Cayley identities.

LaTeX2e, 144 pages. Version 2 has a slightly changed title and abstract, and includes new remarks in Sections 1, 2.6 and (especially) 9 concerning prehomogeneous vector spaces and citing the prior work of Sugiyama (2011). To be published in Advances in Applied Mathematics

References in corpus (2)

• Grassmann Integral Representation for Spanning Hyperforests
• Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities

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