  A polynomial ring with rational coefficients is an irreducible representation
of Lie algebras of endomorphisms of exterior powers of a infinite countable
dimensional $\mathbb{Q}$-vector space. We give an explicit description of it,
using suitable vertex operators on exterior algebras, which mimick those
occurring in the bosonic vertex representation of the Lie algebra $gl_\infty$,
due to Date--Jimbo--Kashiwara and Miwa (DJKM).
