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arXiv · 1105.6270

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

Sergio Caracciolo·Alan D. Sokal·Andrea Sportiello
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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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Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello. 2011-05-31. Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians. https://doi.org/10.1016/j.aam.2012.12.001

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