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

作者：Caracciolo Sergio; Sokal Alan D^*; Sportiello Andrea

来源：Advances in Applied Mathematics, 2013, 50(4): 474-594.

DOI：10.1016/j.aam.2012.12.001

• Cayley identity
• Capelli identity
• Determinant
• Pfaffian
• Bernstein-Sato polynomial
• b-Function
• Prehomogeneous vector space
• Cayley operator
• Omega operator
• Omega process
• Classical invariant theory
• Grassmann algebra
• Exterior algebra
• Grassmann-Berezin integration

The classic Cayley identity states that %26lt;br%26gt;det(partial derivative)(det X)(s) = s(s + 1)...(s + n - 1)(det x)(s-1) %26lt;br%26gt;where X = (x(ij)) is an n x n matrix of indeterminates and partial derivative = (partial derivative/partial derivative x(ij)) 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 %26quot;diagonal-parametrized%26quot; Cayley identities, a pair of %26quot;Laplacian-parametrized%26quot; Cayley identities, and the %26quot;product-parametrized%26quot; and %26quot;border-parametrized%26quot; rectangular Cayley identities.

• 出版日期2013-4