Hasse–Schmidt derivations and Cayley–Hamilton theorem for exterior algebras

Using the natural notion of Hasse–Schmidt derivations on an exterior algebra, we relate two classical and seemingly unrelated subjects. The first is the famous Cayley–Hamilton theorem of linear algebra, “each endomorphism of a finite-dimensional vector space is a root of its own characteristic polynomial”, and the second concerns the expression of the bosonic vertex operators occurring in the representation theory of the (infinite-dimensional) Heisenberg algebra.