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Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and Pfaffians. (English) Zbl 1277.05012

Summary: The classic Cayley identity states that \[ \mathrm{det}(\partial )(\mathrm{det}X)^s=s(s+1)\cdots (s+n-1)(\mathrm{det}X)^{s-1} \] where \(X=(x_{ij})\) is an \(n\times n\) matrix of indeterminates and \(\partial =(\partial /\partial x_{ij})\) 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.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E99 Algebraic combinatorics
11S90 Prehomogeneous vector spaces
13A50 Actions of groups on commutative rings; invariant theory
13N10 Commutative rings of differential operators and their modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
15A15 Determinants, permanents, traces, other special matrix functions
15A23 Factorization of matrices
15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A72 Vector and tensor algebra, theory of invariants
15A75 Exterior algebra, Grassmann algebras
16S32 Rings of differential operators (associative algebraic aspects)
20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
32C38 Sheaves of differential operators and their modules, \(D\)-modules
43A85 Harmonic analysis on homogeneous spaces
81T18 Feynman diagrams
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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References:

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