### 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) |
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Pages (from-to) | 474-594 |

Number of pages | 121 |

Journal | Advances in Applied Mathematics |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2013 |

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### Keywords

- 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
- b-Function

### 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