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

Caracciolo, Sergio; Sportiello, Andrea

Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 1, pp. 1-46.

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), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:

$$\mathrm{col}-\det X\mathrm{col}-\det Y=\langle 0\mid \mathrm{col}-\det (aA+X{(I-a^{†}B)}^{-1}Y)\mid 0\rangle$$

Furthermore, if also $Y$ is a Manin matrix, $[Y_{ij},Y_{kl}]=0$ for $i\ne k$, $j\ne l$

$$\mathrm{col}-\det X\mathrm{col}-\det Y=\int 𝒟(\psi ,\overline{\psi })exp\left(\sum _{k\ge 0}\frac{{\left(\overline{\psi }A\psi \right)}^k}{k+1}\left(\overline{\psi }X{B}^{k}Y\psi \right)\right)$$

Here $\langle 0\mid$ and $\mid 0\rangle$, are respectively the bra and the ket of the ground state, $a^{†}$ and $a$ the creation and annihilation operators of a quantum harmonic oscillator, while ${\overline{\psi }}_i$ and ${\psi }_i$ are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which $A$ and $B$ are null matrices, and of the non-commutative generalization, the Capelli identity, in which $A$ and $B$ are identity matrices and $[X_{ij},X_{k\ell }]=[Y_{ij},Y_{k\ell }]=0$.

Classification : 05-XX, 17-XX
Mots-clés : Invariant Theory, Capelli identity, non-commutative determinant, Lukasiewicz paths, right-quantum matrix, Cartier-Foata matrix, Manin matrix