English

Fischer decomposition for polynomials on superspace

Complex Variables 2015-08-17 v1 Mathematical Physics math.MP Representation Theory

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.

Cite

@article{arxiv.1508.03426,
  title  = {Fischer decomposition for polynomials on superspace},
  author = {Roman Lavicka and Dalibor Smid},
  journal= {arXiv preprint arXiv:1508.03426},
  year   = {2015}
}
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