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

Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type 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.

A bijection proof of the Capelli's identity

In this article, a short combinatorial proof of the Capelli's identity is given. It also leads to an easy proof of the Capelli--Cauchy--Binet identity, a more general form of Capelli's identity. With the technique introduced, the Turnbull's identity can be proven by sum with signs. At the end of the article, the general Cayley identity is discussed as a dual version of Capelli's identity.

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。

Decomplexification of the Capelli identities and holomorphic factorization

The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain matrices with noncommutative entries. They have applications in representation theory and integrable systems. We propose new examples of these identities, constructed according to the following principle. For several known identities for $n$ by $n$ matrices we construct new identity for $2n$ by $2n$ matrices where each element $z$ of the original matrix is substituted by 2x2 matrix of the form $[real(z) ~imag(z); ~ -imag(z) ~ real(z)]$, i.e. we view the original identity as complex valued, while the new identity is it…

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…

Noncommutative multiplicative norm identities for the quaternions and the octonions

We present Capelli type identities associated with the quaternions and the octonions, which are noncommutative versions of multiplicative norm identities for the quaternions and the octonions.