# Hasse--Schmidt Derivations and Cayley--Hamilton Theorem for Exterior Algebras

## TL;DR

This paper explores the connection between Hasse--Schmidt derivations on exterior algebras, the Cayley--Hamilton theorem, and bosonic vertex operators, revealing new links between algebraic and representation-theoretic concepts.

## Contribution

It introduces a novel approach using Hasse--Schmidt derivations to relate classical linear algebra results with infinite-dimensional representation theory.

## Key findings

- Established a link between Cayley--Hamilton theorem and Hasse--Schmidt derivations.
- Connected algebraic derivations with bosonic vertex operators.
- Provided new insights into the structure of exterior algebras and their applications.

## Abstract

Using the natural notion of {\em 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.