Action of free fermions on Symmetric Functions

TL;DR

This paper develops a determinantal formula for endomorphisms of exterior algebras, unifying classical symmetric function identities and bosonic vertex operator representations within a fermionic framework.

Contribution

It introduces a unified determinantal formula for endomorphisms acting on exterior algebras, connecting symmetric functions, Schubert calculus, and Lie algebra representations.

Findings

Recovers Jacobi-Trudy and Giambelli formulas as special cases

Expresses bosonic shadows via determinantal formulas

Links fermionic actions to classical symmetric function theory

Abstract

The Clifford algebra of the endomorphisms of the exterior algebra of a countably dimensional vector space induces natural bosonic shadows, i.e. families of linear maps between the cohomologies of complex grassmannians. The main result of this paper is to provide a determinantal formula expressing generating functions of such endomorphisms unifying several classical special cases. For example the action over a point recovers the Jacobi-Trudy formula in the theory of symmetric functions or the Giambelli's one in classical Schubert calculus, whereas the action of degree preserving endomorphisms take into account the finite type version of the Date-Jimbo-Kashiwara-Miwa bosonic vertex operator representation of the Lie algebra $gl(\infty)$. The fermionic actions on (finite type) bosonic spaces is described in terms of the classical theory of symmetric functions. The main guiding principle is the fact that the exterior algebra is a (non irreducible) representation of the ring of symmetric functions, which is the way we use to spell the ``finite type'' Boson-Fermion correspondence.