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

Abstract

Using the natural notion of Hasse--Schmidt derivations on an exterior algebra}, we relate two classical and seemingly unrelated subjects. The first is the celebrated Cayley--Hamilton theorem of linear algebra, {\em 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) Heinsenberg algebra.