Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities: I. generalizations of the Capelli and Turnbull identities

Sergio Caracciolo; Alan D. Sokal; Andrea Sportiello

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities: I. generalizations of the Capelli and Turnbull identities

Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello

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

Abstract

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.

Original language	English (US)
Pages (from-to)	1-43
Number of pages	43
Journal	Electronic Journal of Combinatorics
Volume	16
Issue number	1
State	Published - Aug 7 2009

Keywords

Capelli identity
Cauchy-binet theorem
Cayley identity
Classical invariant theory
Columndeterminant
Determinant
Noncommutative determinant
Noncommutative ring
Permanent
Representation theory
Row-determinant
Turnbull identity
Weyl algebra

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

Cite this

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title = "Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities: I. generalizations of the Capelli and Turnbull identities",

abstract = "We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.",

keywords = "Capelli identity, Cauchy-binet theorem, Cayley identity, Classical invariant theory, Columndeterminant, Determinant, Noncommutative determinant, Noncommutative ring, Permanent, Representation theory, Row-determinant, Turnbull identity, Weyl algebra",

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

year = "2009",

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AU - Sokal, Alan D.

AU - Sportiello, Andrea

PY - 2009/8/7

Y1 - 2009/8/7

N2 - We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.

AB - We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.

KW - Capelli identity

KW - Cauchy-binet theorem

KW - Cayley identity

KW - Classical invariant theory

KW - Columndeterminant

KW - Determinant

KW - Noncommutative determinant

KW - Noncommutative ring

KW - Permanent

KW - Representation theory

KW - Row-determinant

KW - Turnbull identity

KW - Weyl algebra

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JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

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ER -

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities: I. generalizations of the Capelli and Turnbull identities

Abstract

Keywords

ASJC Scopus subject areas

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