Annales de l’Institut Henri Poincaré D


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Volume 1, Issue 1, 2014, pp. 1–46
DOI: 10.4171/AIHPD/1

Published online: 2014-02-04

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

Sergio Caracciolo[1] and Andrea Sportiello[2]

(1) Università degli Studi di Milano, Italy
(2) Université Paris-Nord, Villetaneuse, France

We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that \[\hbox{$[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$},\] satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds: $$\mathrm {col-det } \ X \ \mathrm { col-det } \ Y \ = \langle 0\mid \mathrm { col-det } \ (aA + X (I-a^{\dagger} B)^{-1} Y)\mid 0\rangle$$ Furthermore, if also $Y$ is a Manin matrix, $[Y_{ij},Y_{kl}]=0$ for $i\neq k$, $j\neq l$ $$\mathrm {col-det } \ X \ \mathrm { col-det } \ Y =\int \mathcal{D}(\psi, \bar{\psi}) \exp \big(\sum_{k \geq 0}\frac{(\bar{\psi} A \psi)^{k}}{k+1}(\bar{\psi} X B^k Y \psi)\big)$$ Here $\langle 0 \mid$ and $\mid 0\rangle$, are respectively the bra and the ket of the ground state, $a^{\dagger}$ and $a$ the creation and annihilation operators of a quantum harmonic oscillator, while $\bar{\psi}_i$ 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 $[X_{ij},X_{k\ell}]=[Y_{ij},Y_{k\ell}]=0$.

Keywords: Invariant Theory, Capelli identity, non-commutative determinant, Lukasiewicz paths, right-quantum matrix, Cartier-Foata matrix, Manin matrix

Caracciolo Sergio, Sportiello Andrea: Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), 1-46. doi: 10.4171/AIHPD/1