Partial differential equation approach to F₄

Abstract

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F₄ over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k≥2 is greater than or equal to 〚k/\mathrm{3}〛+〚(k-\mathrm{2})/\mathrm{3}〛+\mathrm{2}.

Keywords

Lie algebra / F₄-module / singular vector / harmonic polynomial / partial differential equation

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