Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 637-649

ALGEBRA OF INVARIANTS FOR THE RARITA-SCHWINGER OPERATORS

David Eelbode and Dalibor Smíd

Ghent University, Department of Mathematical Analysis
Galglaan 2, 9000 Ghent, Belgium; deef 'at' cage.ugent.be

Charles University, Mathematical Institute
Sokolovska 83, 186 75 Prague 8, Czech Republic; smid 'at' karlin.mff.cuni.cz

Abstract. Rarita-Schwinger operators (or Higher Spin Dirac operators) are generalizations of the Dirac operator to functions valued in representations S_k with higher spin k. Their algebraic and analytic properties are currently studied in Clifford analysis. As a part of this pursuit, we describe the algebra of invariant End S_k-valued polynomials, i.e., the algebra of invariant constant-coefficient differential operators acting on these representations. The main theorem states that this algebra is generated by the powers of the Rarita-Schwinger and Laplace operators. This algebra is the algebraic part of the Howe dual superalgebra corresponding to the Pin group acting on S_k.

2000 Mathematics Subject Classification: Primary 15A66.

Key words: Rarita-Schwinger operators, invariant polynomials, Howe duality, Fischer decomposition, Clifford analysis.

Reference to this article: D. Eelbode and D. Smíd: Algebra of invariants for the Rarita-Schwinger operators. Ann. Acad. Sci. Fenn. Math. 34 (2009), 637-649.

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