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Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I: Generalizations of the Capelli and Turnbull identities. (English) Zbl 1192.15001
The authors prove noncommutative versions of the classical Cauchy-Binet formula. These imply generalizations of the Capelli identity used in classical invariant theory, of Turnbull’s Capelli type identity for symmetric matrices, of Turnbull’s permanental identity for antisymmetric matrices, and of the “Cayley” identity. Proofs consist of simple combinatorial manipulations of commutators. A more intricate combinatorial approach was previously used by {\it D. Foata} and {\it D. Zeilberger} [J. Comb. 1, 49--58 (1994; Zbl 0810.05008)] to reprove the Capelli and Turnbull identities. A combinatorial proof of the Howe-Umeda-Kostant-Sahi antisymmetric (determinantal) analogue of the Capelli identity remains to be found. However, the paper under review proposes a conjectured generalization of the Howe-Umeda-Kostant-Sahi identity for even dimension and a conjectured new version of it for odd dimension.

MSC:
15A15Determinants, permanents, other special matrix functions
05A19Combinatorial identities, bijective combinatorics
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