Fischer decomposition for polynomials on superspace

Abstract

Recently, the Fischer decomposition for polynomials on superspace Rm|2n (that is, polynomials in m commuting and 2n anti-commuting variables) has been obtained unless the superdimension M = m - 2n is even and non-positive. In this case, it turns out that the Fischer decomposition of polynomials into spherical harmonics is quite analogous as in Rm and it is an irreducible decomposition under the natural action of Lie superalgebra osp(m|2n). In this paper, we describe explicitly the Fischer decomposition in the exceptional case when M ∈ -2N0. In particular, we show that, under the action of osp(m|2n), the Fischer decomposition is not, in general, a decomposition into irreducible but just indecomposable pieces.