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

Sergio Caracciolo; Alan D. Sokal; Andrea Sportiello

doi:10.1016/j.aam.2012.12.001

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

Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello

Research output: Contribution to journal › Article › peer-review

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	https://doi.org/10.1016/j.aam.2012.12.001
State	Published - Apr 2013

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

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10.1016/j.aam.2012.12.001

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title = "Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians",

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.",

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",

author = "Sergio Caracciolo and Sokal, {Alan D.} and Andrea Sportiello",

note = "Funding Information: This research was supported in part by U.S. National Science Foundation grants PHY-0099393 and PHY-0424082.",

year = "2013",

month = apr,

doi = "10.1016/j.aam.2012.12.001",

language = "English (US)",

volume = "50",

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AU - Caracciolo, Sergio

AU - Sokal, Alan D.

AU - Sportiello, Andrea

N1 - Funding Information: This research was supported in part by U.S. National Science Foundation grants PHY-0099393 and PHY-0424082.

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 - 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

KW - b-Function

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JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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