A New Multiplicity Formula for the Weyl Modules of Type A

A monomial basis and a filtration of subalgebras for the universal enveloping algebra $U(g_l)$ of a complex simple Lie algebra $g_l$ of type $A_l$ is given in this note. In particular, a new multiplicity formula for the Weyl module $V(\lambda )$ of $U(g_l)$ is obtained in this note.