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Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians (1105.6270v2)

Published 31 May 2011 in math.CO, math-ph, math.AG, and math.MP

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.

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