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Journal of Lie Theory 21 (2011), No. 1, 145--164
Copyright Heldermann Verlag 2011

A Cubic E6-Generalization of the Classical Theorem on Harmonic Polynomials

Xiaoping Xu
Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. Dickson invariant trilinear form is the unique fundamental invariant in the polynomial algebra over the basic irreducible module of E6. In this paper, we prove that the space of homogeneous polynomial solutions with degree $m$ for the dual cubic Dickson invariant differential operator is exactly a direct sum of [m/2]+1 explicitly determined irreducible E6-submodules and the whole polynomial algebra is a free module over the polynomial algebra in the Dickson invariant generated by these solutions. Thus we obtain a cubic E6-generalization of the above classical theorem on harmonic polynomials.

Keywords: Harmonic polynomial, E-6 Lie algebra, irreducible module, Dickson invariant, invariant differential operator, solution space.

MSC: 17B10, 17B25; 17B01

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