Fischer decomposition for polynomials on superspace

Abstract

It is well-known that polynomials decompose into spherical harmonics. This result is called separation of variables or the Fischer decomposition. In the paper we prove the Fischer decomposition for spinor valued polynomials in k vector variables of R m under the stable range condition m ≥ 2k. Here the role of spherical harmonics is played by monogenic polynomials, that is, polynomial solutions of the Dirac equation in k vector variables.1991 Mathematics Subject Classification. 30G35, 17B10.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

“…In [24] the following generalisation of the Fischer decomposition was obtained that still holds for the exceptional case M ∈ −2N.…”

Type-B formal higher spin gravity

“…where now Φ, Ξ belong to the kernel of (5.19). All the steps leading to the system (5.16) remain unchanged except that now one needs to employ the known results [66][67][68] on the structure of polynomial algebras in the supersymmetric case in order to make sure that the ∆ −1 cohomology is concentrated in degree zero in c −− , c 0 , c − -ghosts. In this way one arrives at the system (5.16) but with a, f + belonging to the kernel of (5.19).…”

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$

“…In [19] the following generalisation of the Fischer decomposition was obtained that still holds for the exceptional case M ∈ −2N. Proposition 2.5 (Generalised Fischer decomposition).…”

“…X = (0, L e k , 0), k = 0. In a similar fashion to(19), we find− SB (L x 0k f ) (z) = − exp(−z 0 ) W I 0 (x|z) exp(−2x 0 )L x 0k f (x) = exp(−z 0 ) W L x 0k (I 0 (x|z) exp(−2x 0 )) f (x) = (x 0 z k + z k x 0 ) I 1 (x|z) exp(−z 0 − 2x 0 )f (x) − 2 SB(x k f )(z).…”