Hasse-Schmidt derivations on Grassmann semi-algebras

The talk will be split into two parts. The first will be about the notion of Hasse-Schmidt derivation on a classical exterior algebra, which I introduced years ago to deal with Schubert calculus for complex Grassmannians. In this first part, I will focus on the purely combinatorial features of the construction suited to be transferred in the second part of the talk, concerned with some joint work in progress with Louis Rowen and Adam Chapman. The new framework will be the more general one of Rowen's monoidal triples. We will analyze a few weaker  notions of exterior semi-algebra and how much of the theory discussed in the first part can be extended to this more demanding situation. The kind of results proposed suggests the possibility of extending a classical part of representation theory coming from the theory of infinite-dimensional integrable systems, which will be briefly discussed while highlighting its promising potential.