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 i j , Y k ℓ ] = - A i ℓ B k j ,

satisfying suitable commutation relations (in particular, X is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:

col - det X col - det Y = 〈 0 ∣ col - det ( a A + X ( I - a † B ) - 1 Y ) ∣ 0 〉

Furthermore, if also Y is a Manin matrix, [Y ij ,Y kl ]=0 for i≠k, j≠l

col - det X col - det Y = ∫ 𝒟 ( ψ , ψ ¯ ) exp ∑ k ≥ 0 ( ψ ¯ A ψ ) k k + 1 ( ψ ¯ X B k Y ψ )

Here 〈0∣ and ∣0〉, 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 ψ ¯ i and ψ 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 kℓ ]=[Y ij ,Y kℓ ]=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