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

I. Generalizations of the Capelli and Turnbull identities

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.

Se p 19 93 COMBINATORIAL PROOFS OF CAPELLI ’ S AND TURNBULL ’ S IDENTITIES FROM CLASSICAL INVARIANT THEORY

Factorial Supersymmetric Schur Functions and Super Capelli Identities

AND SUPER CAPELLI IDENTITIES Alexander Molev Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia (e-mail: [email protected]) Abstract A factorial analogue of the supersymmetric Schur functions is introduced. It is shown that factorial versions of the Jacobi{Trudi and Sergeev{Pragacz formulae hold. The results are applied to construct a ...

Transfer of Ideals and Quantization of Small Nilpotent Orbits

We introduce and study a transfer map between ideals of the universal enveloping algebras of two members of a reductive dual pair of Lie algebras. Its definition is motivated by the approach to the real Howe duality through the theory of Capelli identities. We prove that this map provides a lower bound on the annihilators of theta lifts of representations with a fixed annihilator ideal. We also...

A Remark on the Higher Capelli Identities

A simple proof of the higher Capelli identities is given. Mathematics Subject Classifications (1991). 17B10, 17B35.