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

Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory

Capelli's and Turnbull's classical identities are given elegant combinatorial proofs.

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation

We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that $[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$, satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), the following identity holds: $ \mathrm{coldet} X \mathrm{coldet} Y = < 0 | \mathrm{coldet} (a A + X (I-a^{\dagger} B)^{-1} Y) |0 > $. Furthermore, if also $Y$ is a Manin matrix, $ \mathrm{coldet} X \mathrm{coldet} Y =\int \mathcal{D}(ψ, ψ^{\dagger}) \exp [ \sum_{k \geq 0} \frac{1}{k+1} (ψ^{\dagger} A ψ)^{k} (ψ^{\dagger} X B^k Y ψ) ] $. Notations: $ < 0 |$, $| 0 >$, are respec…

一般Capelli型恒等式

General Capelli-type identities

本文提出新方法，推导出包含经典及Okounkov、Williamson恒等式为特例的更一般Capelli型恒等式，并得到广义Turnbull及Howe-Umeda-Kostant-Sahi恒等式，证实了相关猜想。