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

Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello

    Research output: Contribution to journalArticle

    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 languageEnglish (US)
    Pages (from-to)474-594
    Number of pages121
    JournalAdvances in Applied Mathematics
    Volume50
    Issue number4
    DOIs
    StatePublished - 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

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