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

Caracciolo, Sergio; Sportiello, Andrea

doi:10.4171/aihpd/1

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

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Résumé

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, $[Y_{i j}, Y_{k l}] = 0$ for $i \neq k$ , $j \neq l$

col - \det X col - \det Y = \int 𝒟 (ψ, \bar{ψ}) exp (\sum_{k \geq 0} \frac{{(\bar{ψ} A ψ)}^{k}}{k + 1} (\bar{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 ${\bar{ψ}}_{i}$ 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 $[X_{i j}, X_{k ℓ}] = [Y_{i j}, Y_{k ℓ}] = 0$ .

Publié le : 2014-02-04

MR Zbl

DOI : 10.4171/aihpd/1

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

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     author = {Caracciolo, Sergio and Sportiello, Andrea},
     title = {Noncommutative determinants, {Cauchy{\textendash}Binet} formulae, and {Capelli-type}  identities {II.} {Grassmann} and quantum oscillator algebra representation},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {1--46},
     volume = {1},
     number = {1},
     year = {2014},
     doi = {10.4171/aihpd/1},
     mrnumber = {3166201},
     zbl = {1383.15007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpd/1/}
}

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Caracciolo, Sergio; Sportiello, Andrea. Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type  identities II. Grassmann and quantum oscillator algebra representation. Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 1, pp. 1-46. doi: 10.4171/aihpd/1

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