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 2013

    Fingerprint

    Pfaffian
    Cayley
    Algebra
    Determinant
    Derivatives
    Derivative
    Grassmann Algebra
    Exterior Algebra
    Partial derivative

    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

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

    In: Advances in Applied Mathematics, Vol. 50, No. 4, 04.2013, p. 474-594.

    Research output: Contribution to journalArticle

    Caracciolo, Sergio ; Sokal, Alan D. ; Sportiello, Andrea. / Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians. In: Advances in Applied Mathematics. 2013 ; Vol. 50, No. 4. pp. 474-594.
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