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Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I: Generalizations of the Capelli and Turnbull identities
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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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.
DOI : 10.37236/192
Classification : 15A15, 05A19
Mots-clés : determinant, noncommutative determinant, row-determinant, column-determinant, Cauchy-Binet theorem, permanent, noncommutative ring, Capelli identity, Turnbull identity, Cayley identity, classical invariant theory, representation theory, Weyl algebra, right-quantum matrix, Cartier-Foata matrix, Manin matrix, Howe-Umeda-Kostant-Sahi identity 
@article{10_37236_192,
     author = {Sergio Caracciolo and Alan D. Sokal and Andrea Sportiello},
     title = {Noncommutative determinants, {Cauchy-Binet} formulae, and {Capelli-type} identities. {I:} {Generalizations} of the {Capelli} and {Turnbull} identities},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/192},
     zbl = {1192.15001},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/192/}
}
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Sergio Caracciolo; Alan D. Sokal; Andrea Sportiello. Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I: Generalizations of the Capelli and Turnbull identities. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/192

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