Combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

Details Combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians Combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

2011-05-31

arxiv.org/abs/1105.6270v1

Mathematics

Combinatorics

LaTeX2e, 143 pages

Scientific paper

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 the two-matrix and multi-matrix rectangular Cayley identities, the one-matrix rectangular antisymmetric Cayley identity, 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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