4. Hasse–Schmidt Derivations on Exterior Algebras

We are going to keep the same notation of Chapter 3, unless otherwise specified. The main goal of this chapter is to introduce the pivotal algebraic notion of Hasse–Schmidt (HS) derivation on an exterior algebra and develop its basic formalism, with emphasis on what we call integration by parts. The latter is a key tool and will be used heavily throughout. The original motivation comes from Schubert calculus (see, e.g. Chapter 5), a subject concerned with the intersection theory on Grassmann varieties, quickly revised in Section 5.​4 HS-derivations on Grassmann algebras also provide a natural framework to state and prove a generalization (as in [51]) of the classical celebrated Cayley–Hamilton theorem: any endomorphism is a root of its characteristic polynomial. This is shown in Section 4.2 and will be used in Chapter 5 to construct a structure of free B _r -module out of any free abelian group of rank n  ≥ r. That construction will enable us to describe the cone of decomposable tensors in an exterior power by means of the same compact and elegant formula we advertised in the introduction. Section 4.3 offers a few simple applications to matrix exponentials and first integrals of linear ODEs with constant coefficients, by means of the same language of Section 2.​5