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

TL;DR

This paper provides combinatorial proofs, including Grassmann algebra methods, for various Cayley-type identities involving derivatives of determinants and pfaffians, introducing new parametrized identities.

Contribution

It introduces new Cayley-type identities and offers straightforward combinatorial proofs, including Grassmann algebra techniques, for classical and novel identities.

Findings

Proved new diagonal-parametrized Cayley identities

Established Laplacian-parametrized Cayley identities

Derived product- and border-parametrized rectangular Cayley identities

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.