Schubert Calculus and the Heisenberg Algebra

TL;DR

This paper establishes a novel connection between the cohomology of the infinite flag variety, infinite permutations, and the Heisenberg algebra, using back-stable Schubert polynomials and a fermionic model.

Contribution

It introduces a fermionic model for infinite permutations and relates pipedreams to Hamiltonian evolution in this framework.

Findings

Hilbert space with infinite permutations forms a Heisenberg algebra representation

Back-stable Schubert polynomials provide an isomorphism between different models

Pipedreams correspond to fermionic Hamiltonian evolution

Abstract

We show that the Hilbert space with basis indexed by infinite permutations and the cohomology ring of the infinite flag variety can be seen as representations of the Heisenberg algebra, which are isomorphic using the back-stable Schubert polynomials. We give a model for infinite permutations as certain two dimensional fermions, generalizing the Maya diagram construction for partitions. Under this framework, the pipedream model for Schubert polynomials can be viewed as the Hamiltonian time evolution of the 2D fermions.