Frontiers of Mathematics in China >

2011 , Vol. 6 >Issue 4: 759 - 774

DOI: https://doi.org/10.1007/s11464-011-0131-z

RESEARCH ARTICLE

Partial differential equation approach to F4

  • Xiaoping XU
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  • Hua Loo-Keng Key Mathematical Laboratory, Institute of Mathematics, Academy of Mathematics & System Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received date: 26 Aug 2010

Accepted date: 29 Mar 2011

Published date: 01 Aug 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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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/3〛+〚(k-2)/3〛+2.

Key words： Lie algebra; F4-module; singular vector; harmonic polynomial; partial differential equation

Cite this article

Xiaoping XU . Partial differential equation approach to F4[J]. Frontiers of Mathematics in China, 2011 , 6(4) : 759 -774 . DOI: 10.1007/s11464-011-0131-z

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