Fischer decomposition for polynomials on superspace

2015/08/14 by Roman Lávička, Lavicka, Roman, Dalibor Šmíd +1 · 1 citation
Mathematics · Physics and Astronomy · #17B10 #30G35 #58C50 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1508.03426

openalex publication_date 2015/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Howe duality for the dual pair (SpO(2n|1) , \mathfrakosp(2|2))

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