Fischer decomposition for polynomials on superspace

TL;DR

This paper explores the Fischer decomposition for polynomials on superspace, especially focusing on the exceptional case where the superdimension is even and non-positive, revealing indecomposable but not irreducible structures under osp(m|2n).

Contribution

It explicitly describes the Fischer decomposition in the exceptional superdimension case, highlighting the indecomposable nature of the decomposition under osp(m|2n).

Findings

Fischer decomposition is analogous to classical case when superdimension is even and non-positive.

Decomposition into spherical harmonics is not always irreducible in the exceptional case.

The structure under osp(m|2n) can be indecomposable but not irreducible.

Abstract

Recently, the Fischer decomposition for polynomials on superspace R^{m|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 R^m 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 is even and non-positive. In particular, we show that, under the action of osp(m|2n), the Fischer decomposition is not, in general, a decomposition into irreducible but indecomposable pieces.