Polynomial Representation of $F_4$ and a New Combinatorial Identity about Twenty-Four

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 $F_4$ 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 $k\geq 2$ is $\geq [|k/3|]+[|(k-2)/3|]+2$.