# A DG-extension of symmetric functions arising from higher representation theory

## TL;DR

This paper extends symmetric functions via a higher representation theory approach, connecting algebraic structures to the cohomology of Grassmannians and introducing new differentials that relate to equivariant cohomology.

## Contribution

It introduces a new DG-extension of symmetric functions from higher representation theory, linking algebraic structures to geometric cohomology.

## Key findings

- Extended symmetric functions form a subalgebra of polynomial and exterior algebra
- The algebra admits differentials making it a sub-DG-algebra of the extended nilHecke algebra
- The ring with differentials is quasi-isomorphic to Grassmannian cohomology

## Abstract

We investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to $GL(N)$-equivariant cohomology of Grassmannians.