### Abstract

The classic Cayley identity states thatdet(∂)(^{detX)s}=s(s+1) ...(s+n-1)(^{detX)s-1} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 474-594 |

Number of pages | 121 |

Journal | Advances in Applied Mathematics |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2013 |

### Fingerprint

### Keywords

- 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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Advances in Applied Mathematics*,

*50*(4), 474-594. https://doi.org/10.1016/j.aam.2012.12.001

**Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians.** / Caracciolo, Sergio; Sokal, Alan D.; Sportiello, Andrea.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 50, no. 4, pp. 474-594. https://doi.org/10.1016/j.aam.2012.12.001

}

TY - JOUR

T1 - Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

AU - Caracciolo, Sergio

AU - Sokal, Alan D.

AU - Sportiello, Andrea

PY - 2013/4

Y1 - 2013/4

N2 - The classic Cayley identity states thatdet(∂)(detX)s=s(s+1) ...(s+n-1)(detX)s-1 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.

AB - The classic Cayley identity states thatdet(∂)(detX)s=s(s+1) ...(s+n-1)(detX)s-1 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.

KW - b-Function

KW - Bernstein-Sato polynomial

KW - Capelli identity

KW - Cayley identity

KW - Cayley operator

KW - Classical invariant theory

KW - Determinant

KW - Exterior algebra

KW - Grassmann algebra

KW - Grassmann-Berezin integration

KW - Omega operator

KW - Omega process

KW - Pfaffian

KW - Prehomogeneous vector space

UR - http://www.scopus.com/inward/record.url?scp=84875226861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875226861&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2012.12.001

DO - 10.1016/j.aam.2012.12.001

M3 - Article

AN - SCOPUS:84875226861

VL - 50

SP - 474

EP - 594

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 4

ER -