Partial differential equation approach to F ₄

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 F ₄ 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.

Keywords

Lie algebra / F ₄-module / singular vector / harmonic polynomial / partial differential equation

References

[1] Adams J. Lectures on Exceptional Lie Groups, 1996, London: The University of Chicago Press Ltd.

[2] Brion M. Invariants d’un sous-groupe unipotent maxaimal d’un groupe semi-simple. Ann Inst Fourier (Grenoble), 1983, 33, 1-27

[3] Garland H., Lepowsky J. Lie algebra homology and the Macdonald-Kac formulas. Invent Math, 1976, 34, 37-76

[4] Humphreys J. E. Introduction to Lie Algebras and Representation Theory, 1972, New York: Springer-Verlag.

[5] Kac V. Infinite-Dimensional Lie Algebras, 1982, Boston: Birkhäuser.

[6] Kostant B. On Macdonald’s η-function formula, the Laplacian and generalized exponents. Adv Math, 1976, 20, 179-212

[7] Kostant B. Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra. Invent Math, 2004, 158, 181-226

[8] Lepowsky J., Wilson R. A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv Math, 1982, 45, 21-72

[9] Lepowsky J., Wilson R. The structure of standard modules, I: universal algebras and the Rogers-Ramanujan identities. Invent Math, 1984, 77, 199-290

[10] Macdonald I. Affine root systems and Dedekind’s η-function. Invent Math, 1972, 15, 91-143

[11] Xu X. Kac-Moody Algebras and Their Representations, 2007, Beijing: Science Press.