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.

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians / S. Caracciolo, A.D. Sokal, A. Sportiello. - In: ADVANCES IN APPLIED MATHEMATICS. - ISSN 0196-8858. - 50:4(2013), pp. 474-594. [10.1016/j.aam.2012.12.001]

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

S. Caracciolo;A. Sportiello
2013

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 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.
b-Function; Bernstein-Sato polynomial; Capelli identity; Cayley identity; Cayley operator; Classical invariant theory; Determinant; Exterior algebra; Grassmann algebra; Grassmann-Berezin integration; Omega operator; Omega process; Pfaffian; Prehomogeneous vector space
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
ADVANCES IN APPLIED MATHEMATICS
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/218958
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