Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians |
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Authors: | Sergio Caracciolo Alan D Sokal Andrea Sportiello |
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Institution: | 1. Dipartimento di Fisica and INFN, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy;2. Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA |
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Abstract: | The classic Cayley identity states that 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. |
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Keywords: | primary 05A19 secondary 05E15 05E99 11S90 13A50 13N10 14F10 15A15 15A23 15A24 15A33 15A72 15A75 16S32 20G05 20G20 32C38 43A85 81T18 82B20 |
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