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

Abstract

In memory of those participants of the Voronezh Winter Mathematical School who have already passed into another world where all problems are solved 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 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) Heinsenberg algebra.2010 Mathematics Subject Classification. Primary 15A75, 17B69. 050E5.

The cohomology of the Grassmannian is a gln-module

“…As remarked in [7], the second of ( 10) is the generalization (holding also for free A-module of infinite rank) of the Cayley-Hamilton theorem.…”

“…where E(z, w) n = 0 i,j<n E i,j z i w −j . Equality (7) means that to describe the gl n (Z)-action on an element of B r,n is sufficient to set to zero all the h j with j > n − r that may possibly occur in the expression obtained for n = ∞ using (4). For example, by applying the described recipe, it is easy to see that…”

Schubert Derivations on the Infinite Wedge Power

“…We should finally remark that many of the tools employed in this paper within the framework of Schubert derivations have already been reviewed in other contributions (e.g. [6,7,9,10,11]), which we might well refer to. However, since the vocabulary of HS-derivations is not yet standard, it seems motivated to recall the basic notions and facts without saying anything more about proofs or the self-containedness of this first draft.…”

Bosonic and fermionic representations of endomorphisms of exterior algebras

“…32 and 34] and also [21]). To follow the reference [20, Theorem 6.1] more closely, and be more in line with the subject of the paper, we call (0.2) the boson-fermion correspondence, which has also been considered from the point of view of linear ODEs in [15,10].…”