Polynomial Representation of F₄ and a New Combinatorial Identity about Twenty-Four

2008/10/26 by Xiaoping Xu, Xu, Xiaoping · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Coding theory and cryptography #FOS: Mathematics #Representation Theory (math.RT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.0810.4670

openalex publication_date 2008/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 find all the singular vectors of the polynomial representation of the simple Lie algebra of type F₄ over its basic irreducible module. As applications, we obtain a new combinatorial identity about the number 24 and explicit generators of invariants. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k≥ 2 is ≥ [|k/3|]+[|(k-2)/3|]+2.

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• Representations of Lie Algebras and Coding Theory

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